LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
LCP 11: Part II
Spacewatch
Fig. 1 The Spacewatch 35-inch (0.9-meter) telescope (left) on beautiful Kitt Peak during the winter.
The new telescope will be used exclusively for asteroid searches, relocating objects that have become, quite
literally, lost in space, and for keeping an eye on the whereabouts of newly found objects.
Asteroid and comet collisions became a popular topic in the 1990s. Public awareness of the potential for a
collision with a rogue asteroid or a comet was raised because of several recent occurrences that were all highly
publicised. Many things have contributed to a more collective and political awareness of the dangers of asteroid
collisions: from the Alvarez’ team promotion of the “dinosaur extinction by impact”, the reports of a “near miss”
by 1889 FC asteroid, the Jovian impact of SL-9.that could be watched on TV in real time, to the premature
announcement that the Asteroid 1997XF11 was on a collision course with Earth in 2028. Clearly, the movies
Armageddon and Deep Impact have also contributed to capturing the imagination and the concern of the general
public. As early as 1990 the U.S. House of Representatives, in its NASA Multiyear Authorization Act stated in
part:
The Committee believes that it is imperative that the detection rate of Earth-orbit-crossing
asteroids must be increased substantially, and that the means to destroy or alter the orbits of
asteroids when they threaten collision should be defined and agreed upon internationally.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
Spacewatch is the name of a group at the University of Arizona/s Lunar and Planetary Laboratory (LPL)
The primary goal of this group is
.... the study of statistics of comets and asteroids in order to investigate the collisional evolution of the
solar system’ the discovery of target asteroids for space missions, such as Clementine, whose aim is the
exploration and the protection the Earth from asteroid impact
The Spacewatch program was the result of action taken by Congress in the US that asked for national
workshops in 1990. The astronomer and asteroid expert, Tom Gehrels, of the University of Arizona, started
systematically looking for asteroids using a special telescope, using the newly developed CCD’s (chargecoupled devices) which are superior to photographic emulsions.
In 1992 a NASA study the Spaceguard Report, partly an outgrowth of the Spacewatch program, showed
that all potential Earth impactors down to a 1 kilometer in size could be discovered and tracked in a program
costing only $300 million, spread over 25 years. As we have seen, 1 km in diameter is the size of an impactor
that would cause mass destruction, killing about 25-50% of mankind. Astronomers argue that this is a small
investment for buying security against a potentially global destructive agent. Two committees were struck: The
Detection Committee and the Interception Committee. The first covered the astronomical aspects of the problem:
How do you search these objects out and then determine their orbits with sufficient precision such
that any potential impact in the foreseeable future might be identified?
The second committee’s task was to find an answer to the question:
How hard must we push the errant asteroid in order to make it miss the Earth?”
Or more realistically:
How do you shove a 1 kilometer asteroid with a mass of more than one billion tons, and travelling at 30
km/s, out of the way of the Earth? and
“To deliver the appropriate push, what sort of explosive should we use?”
Using nuclear energy seems the obvious answer, but let us consider other sources of energy. One kilogram of
TNT could be represented as the maximum chemical energy we could deliver per unit mass, or 4.1 x 106 Joules.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
It is easy to show (see problem below) that a 1 kilogram mass travelling with a speed of 2.88 km/s has the same
kinetic energy as the chemical energy available from one kilogram of TNT. Therefore, if a ton projectile of TNT
were to hit an asteroid, 90 % of the total energy would come from the kinetic energy of the mass and only 10%
from the ch chemical explosion! It looks like chemical weapons would be unsuitable for asteroid deflection.
Thus, the options we have are:
1.
Large kinetic interceptor, with a mass of at least 10, 000 tons, and
2.
An interceptor with a nuclear warhead.
Granted, we need only a modest change in speed, maybe only about 10 cm/s, but to accomplish that
requires a large amount of energy. For example, to change the speed of a 1 kilometer asteroid by 10 cm/s would
require about 4 x 10 15 of energy, or about 1 megaton of TNT. This might convince us that a comparatively small
weapon would do the job. Unfortunately, a great deal of the energy would be inelastically absorbed by the
asteroid, and only a small amount left over to change its speed. This problem was investigated by Thomas
Ahrens and Alan Harris of the California Institute of Technology.
They considered al sizes of bolides of diameters 0.1, 1, and 10 km. The smallest is the minimal size to cause
significant damage, and the largest is about the size of the biggest Earth-crossing asteroids and periodic
comets. Comets smaller than 0.1 km (100m) could be diverted by kinetic means, but the larges ones would
require alternative methods. The following were suggest
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
1.
Land a device or mechanism on the asteroid or comet that would throw off material (a mass
driver) continuously and at a very high speed,
2.
Choose from various methods of deploying nuclear charges:
a.
Burying nuclear explosives in the center of the body which would blow it
apart.
b.
Deposit charges on the surface so that the explosion would produce a recoil force that in
turn would change the velocity of the asteroid or comet.
c.
Use an explosion about 100 m above the bolide’ s surface to cause a skin layer
to be
ejected that would then produce a recoil force.
Let us consider these in turn. The first approach, blowing up an asteroid, sounds very dramatic but it
really is not the preferred way to divert it. Many large fragments would be produced with unpredictable
trajectories. Digging deep into an asteroid may present unforeseen technological problems. We must also
remember that the centre of mass of the individual parts of the asteroid after explosion will always travel along
the original orbit.
The second idea is more attractive, provided the speed with which the mass is ejected is greater than the
the escape velocity of that asteroid. We have already calculated the escape velocityfrom various sizes of planets,
planetoids and asteroid, so we know that for asteroids of 1 - 30 km across , these are of the order of 1-20 m/s.
The problem is, however, that we still do not know enough about the composition of the asteroid. The explosion
may just fragment the asteroid rather than give it a desired velocity change.
The third approach is the one the researchers recommended. Exploding a nuclear weapon at roughly 0.40
times the diameter of the asteroid, about 30% of its surface would be “bathed” in the neutron radiation of the
explosion. They think that the sudden heating of the surface to a high temperature would produce the ejection of
material above the escape velocity (of that particular asteroid). They found that the required explosive energy is
from about 100 kilotons of TNT for a 1 km asteroid to about 10 megatons for a 10 km body. Using our present
nuclear capabilities we should be able to meet these requirements.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
There are, of course, also suggestions that are quite exotic. These include:
a.
Attaching a giant “solar sail” to provide a small but continuing impulse that would
push the
asteroid out of a collision course.
b.
Alternatively, using a very large sail as a solar collector to focus Sunlight onto the
asteroid surface in an effort to evaporate material that would then produce a jet force as
the gas expanded away from the asteroid.
The first suggestion must be immediately rejected because most asteroids spin and
tumble so that it would be impossible to attach a sail. The second one, however, seems possible.
They showed that a sail of 0.5 km in diameter could deflect an asteroid up to 2 km in size,
assuming continuous operation for a year. The reflective material would only have a mass of
about one ton. Already in 1967 there were groups of scientists concerned about a possible
asteroid collision with the Earth. At MIT a group of engineering students was given the problem
of diverting the asteroid Icarus, which was imagined to be on a collision course with Earth. The
conclusion of the students was that the asteroid could be diverted by using a Saturn V heavy-lift
capability (then available for the Apollo program) and six 100 megaton hydrogen bombs. The
scenario was based on the assumption that there was only about a year left before impact, and
that meant high velocity changes were required.
Icarus is a very interesting asteroid. It was discovered accidentally in 1949 by the famous
American astronomer Walter Baade, using the new 48 inch Schmidt telescope on Palomar Mountain.
The asteroid was about 1 km across, is an Earth-crossing asteroid and has a high eccentricity. In 1949
it was the closest any observed asteroid came to the Earth: about 6 million kilometers.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
Fig. 2 The asteroid Icarus, though only a few hundred meters across, orbits the sun like the planets. its
period is 410 day. What is its mean distance from the sun?
Deflection calculations
A surprising result of orbit deflection calculations is that it takes a very small velocity change to alter the orbit of
an asteroid significantly, if we have sufficient time before the predicted collision. The following example will
illustrate this:
Or more realistically:
How do you shove a 1 kilometer asteroid with a mass of more than one billion tons, and travelling at 30
km/s, out of the way of the Earth?
The CCD ( charge-coupled devices) scanning observations are conducted 20 nights each “lunation” with the
Stewart Observatory 0.9 m Spacewatch Telescope on Kitt Peak Using these data, the organization:
Studies the orbital element distribution of Trojan and Main-Belt asteroids
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
First of all, it must be made clear that predicting an orbit of a comet or asteroid beyond 100
years is tricky and beyond 200 years almost impossible. The best that can said is something like:
“Asteroid X will pass somewhere within five times the radius of the Earth, or that asteroid X has about
a 4% chance of hitting the Earth This information is very important because if you make the decision to
intercept a 1 km asteroid or comet that is predicted to pass within 5 Earth radii, then you must be able
to deflect that object by much more than that amount.
Where would you have to intercept the object to have the greatest effect? Most people seem to
think that pushing an asteroid sideways would be the best thing to do. Surprisingly, the most effective
thing to do is to give it a push along the trajectory in which it is already moving. But the push (impulse)
can be applied either to the trailing end or the leading end, speeding it up or slowing it down. This will
be illustrated in the problems below.
Fig. 3 Deflecting an asteroid using mirrors and the pressure of light.
To deliver the appropriate push, what sort of explosive should we use? Using nuclear energy seems the
obvious answer, but let us consider other sources of energy. One kilogram of TNT could be represented as the
maximum chemical energy we could deliver per unit mass, or 4.1 x 106 Joules. It is easy to show (see problem
below) that a 1 kilogram mass travelling with a speed of 2.88 km/s has the same kinetic energy as the chemical
7
LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
energy available from one kilogram of TNT. Therefore, if a ton projectile of TNT were to hit an asteroid, 90 %
of the total energy would come from the kinetic energy of the mass and only 10% from the chemical explosion!
It looks like chemical weapons would be unsuitable for asteroid deflection.
Thus, the options we have are:
1.
Large kinetic interceptor, with a mass of at least 10, 000 tons, and
2.
An interceptor with a nuclear warhead.
Granted, we need only a modest change in speed, maybe only about 10 cm/s, but to accomplish that
requires a large amount of energy. For example, to change the speed of a 1 kilometer asteroid by 10 cm/s would
require about 4 x 10 15 of energy, or about 1 megaton of TNT. This might convince us that a comparatively small
weapon would do the job. Unfortunately, a great deal of the energy would be inelastically absorbed by the
asteroid, and only a small amount left over to change its speed. This problem was investigated by Thomas
Ahrens and Alan Harris of the California Institute of Technology.
They considered al sizes of bolides of diameters 0.1, 1, and 10 km. The smallest is the minimal size to
cause significant damage, and the largest is about the size of the biggest Earth-crossing asteroids and periodic
comets. Comets smaller than 0.1 km (100m) could be diverted by kinetic means, but the larges ones would
require alternative methods. The following were suggested:
1.
Land a device or mechanism on the asteroid or comet that would throw off material (a mass
driver) continuously and at a very high speed,
2.
Choose from various methods of deploying nuclear charges:
a.
Burying nuclear explosives in the center of the body which would blow it apart.
b.
Deposit charges on the surface so that the explosion would produce a recoil
force that in turn would change the velocity of the asteroid or comet.
c.
Use an explosion about 100 m above the bolide’ s surface to cause a skin layer
to be ejected that would then produce a recoil force
Let us consider these in turn. The first approach, blowing up an asteroid, sounds very dramatic but it
really is not the preferred way to divert it. Many large fragments would be produced with unpredictable
trajectories. Digging deep into an asteroid may present unforeseen technological problems. We must also
remember that the centre of mass of the individual parts of the asteroid after explosion will always
travel along the original orbit.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
The second idea is more attractive, provided the speed with which the mass is ejected is greater than the
the escape velocity of that asteroid. We have already calculated the escape velocity from various sizes of
planets, planetoids and asteroid, so we know that for asteroids of 1 - 30 km across , these are of the order of 1-20
m/s. The problem is, however, that we still do not know enough about the composition of the asteroid. The
explosion may just fragment the asteroid rather than give it a desired velocity change.
The third approach is the one the researchers recommended. Exploding a nuclear weapon at roughly 0.40
times the diameter of the asteroid, about 30% of its surface would be “bathed” in the neutron radiation of the
explosion. They think that the sudden heating of the surface to a high temperature would produce the ejection of
material above the escape velocity (of that particular asteroid). They found that the required explosive energy is
from about 100
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
kilotons of TNT for a 1 km asteroid to about 10 megatons for a 10 km body. Using our present nuclear
capabilities we should be able to meet these requirements.
There are, of course, also suggestions that are quite exotic. These include:
a.
Attaching a giant “solar sail”10
to provide a small but continuing impulse that
would push the asteroid out of a collision course.
b.
Alternatively, using a very large sail as a solar collector to focus Sunlight onto
the asteroid surface in an effort to evaporate material that would then
produce a jet force as the gas expanded away from the asteroid.
The first suggestion must be immediately rejected because most asteroids spin and tumble so that it
would be impossible to attach a sail. The second one, however, seems possible. They showed that a sail of 0.5
km in diameter could deflect an asteroid up to 2 km in size, assuming continuous operation for a year. The
reflective material would only have a mass of about one ton.
Already in 1967 there were groups of scientists concerned about a possible asteroid collision with the
Earth. At MIT a group of engineering students was given the problem of diverting the asteroid Icarus, which was
imagined to be on a collision course with Earth. The conclusion of the students was that the asteroid could be
diverted by using a Saturn V heavy-lift capability (then available for the Apollo program) and six 100 megaton
hydrogen bombs. The scenario was based on the assumption that there was only about a year left before impact,
and that meant high velocity changes were required.
Icarus was discovered accidentally in 1949 by the famous American astronomer Walter Baade, using the
new 48 inch Schmidt telescope on Palomar Mountain. The asteroid was about 1 km across, is an Earth-crossing
asteroid and has a high eccentricity. In 1949 it was the closest any observed asteroid came to the Earth: about 6
million kilometers.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
11
Fig. 4 Deflecting an asteroid.
IL **** An excellent source of videos on asteroid defence
http://planetarydefense.blogspot.com/2007_12_01_archive.html
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
Deflection calculations: Examples
IL *** Discussion of deflecting asteroids
http://pdf.aiaa.org/preview/CDReadyMPDC04_865/PV2004_1433.pdf
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A surprising result of orbit deflection calculations is that it takes a very small velocity change to alter the orbit of
an asteroid significantly, if we have sufficient time before the predicted collision. The following example will
illustrate this:
Imagine a 1 km asteroid, asteroid X 2001 (density about 3.0 cm 3) in a collision course with the Earth.
The asteroid is found to have a perihelion of near 1 AU and aphelion at 4 AU. It is expected that the
asteroid will pass within 5 Earth radii unless the asteroid is diverted.
Initial calculations:
a.
Sketch the elliptical orbit of the asteroid.
b.
Show that the semi-major axis is 2.5 AU.
c.
What is the eccentricity of the asteroid?
d.
Using the vis-viva equation show that the speed of the asteroid (relative to the Sun) at
perihelion will be 37.6 km/s a
e.
Now calculate, using Kepler’s second law, the speed at aphelion.
The effect of changing the velocity by 10 cm/s along the line of the trajectory
1
Give an argument that the change of velocity at perihelion will have a greater effect in changing
the elliptical orbit than changing it (by the same amount) at aphelio
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
2.
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Imagine that asteroid X is nudged by a large, 1000 ton projectile which hits the asteroid in front,
with a contact speed of 30 km/s. The projectile is imbedded in the asteroid.
3.
What will be the change in velocity of the asteroid?
4.
Show that the change in the semi-major axis will be slight, from 2.500000 to 2.5000531 AU., a
distance of about 8000 km. But this is a little more than the radius of the Earth, so how can it
improve our collision probability?
Actually, the asteroid will miss the Earth by much more than 8000 km. Remember, the change
of the semi-major axis will increase the orbital period by a significant amount.
5.
Sow that the orbital period increase of the asteroid is about 66 minutes.
6.
It is now easy to calculate the distance the Earth travels during this time. Calculate this
distance.
7.
Approximately how many Moon radii away will the asteroid pass us?
8.
Debate the following conclusion: “Intercepting dangerous objects is best done near perihelion,
either by increasing or decreasing orbital speed by at least about 10 cm/s”.
Fig. 5 Deflecting an asteroid
On 4 July 2005, NASA collided a projectile with comet Tempel 1 for scientific reasons. One day in the
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
future, scientists anticipate having to do this on a much bigger scale in order to actually shove a celestial
body off its collision course with Earth. Image courtesy: NASA/JPL-Caltech/UMD.
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Read and discuss the following two exerpts taken from:
IL *** Collision scenarios
http://www.innovations-report.de/html/berichte/physik_astronomie/bericht-49763.html
and
IL *** Collision scenarios
http://www.spacedaily.com/reports/Deflecting_Asteroids_Difficult_But_Possible.html
I. The UK’s first engineering feasibility study into missions for deflecting asteroids has begun.
The Engineering and Physical Sciences Research Council (EPSRC) is funding a new three-year study into
interception and deflection strategies for asteroids found to be on a collision course with Earth. Although there
have been similar studies in the past, Dr Gianmarco Radice, department of Aerospace Engineering, University of
Glasgow, and Professor Colin McInnes, department of Mechanical Engineering, University of Strathclyde, are
approaching the subject in a new way
We will be looking at this as engineers. So we want to investigate the practicality of different deflection
strategies,” says McInnes. In other words, it is no use having a brilliant deflection scheme if no one can build it
with current technology.
Although Hollywood blockbusters have popularised the idea of using nuclear weapons to blow up asteroids, the
study will investigate more realistic alternatives such as space mirrors. These would be angled to focus sunlight
onto the incoming object. The intense heat would boil away a section of the asteroid, creating a natural rocket
that pushes the asteroid in the opposite direction. The study will also look into high-speed collisions to literally
knock an asteroid out of the way using no explosives, just a ‘battering ram’ spacecraft.
Asteroids have widely differing compositions, ranging from pure rock or even metal to ice and snow. Knowing
what an asteroid is made from, and therefore its likely strength, is the crucial first step in determining the best
way to divert it without shattering it. “One of the main objectives of this study is to try to associate a particular
deflection strategy with a particular type of asteroid that has to be deviated,” says Radice.
The internal arrangement of Near Earth Objects (NEOs) can critically affect the deviation strategy. Some
asteroids, known as rubble piles, are not solid slabs of rock but loose assemblages. Slamming an object into a
rubble pile would not be very effective in altering its course, because the rubble would absorb the energy of
impact rather like a crumple zone on a car absorbs a crash. Instead, scenarios which melt part of the surface, such
as space mirrors, producing jets of gas that gradually ease the object into a new orbit, are favoured.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
Yet this is about more than just diverting asteroids, no matter how critical that need may one day become. The
biggest part of the study concerns how to intercept such targets. In conventional space exploration, everything is
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precisely worked out beforehand and targets are chosen that have well-known orbits. That’s how NASA recently
bulls-eyed comet Tempel 1 with its Deep Impact mission.
However, a dangerous object is likely to be newly discovered and that means its orbit will be poorly known.
“We’d probably have to launch a deflection mission without a clear idea of where we’re aiming,” says McInnes.
So, the study will seek to find the best strategies for launching space missions into approximate intercept orbits
that can be adjusted later.
To do this, it will investigate the additional fuel that such a spacecraft would require. Because fuel is heavy,
spacecraft are traditionally designed to carry little extra. That will have to change with this new approach to
space exploration.
Such seat-of-the-pants flying could result in more versatile spacecraft across the board. These would be better
able to respond to a variety of unexpected situations. As well as fuel considerations, the team will investigate
‘general purpose’ orbits and flexible navigation strategies that keep a spacecraft’s options open for longer, before
committing it to a final destination.
Natasha Richardson | Quelle: alphagalileo
Weitere Informationen: www.epsrc.ac.uk
Fig. 6 Col. Gen. Vladimir Popovkin, commander of the Russian Military Space Forces, told a
news conference Friday that the national satellite network lacked a spacecraft capable of
preventing an asteroid strike:
Over the last few decades there has been a great deal of debate about the level of danger posed by impacts from
asteroids and comets. It appears the world needs to take the threat of asteroid strikes a lot more seriously.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
Astronomers have already spotted about 800 asteroids, solid rocky celestial bodies, with a diameter of over 1,000
meters (3,250 feet) moving along circumsolar elliptical orbits. However, there may be as many as 2,000 large
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asteroids, and some 135,000 rocks with a diameter of 100 meters (325 feet) and more.
It should be noted that asteroid orbits are unstable and tend to change under the influence of gravitational fields
of the terrestrial planets - Mercury, Venus, Earth and Mars.
An asteroid, which flashed past our planet at a distance of 5 million kilometers (3.1 million miles) in November
1996, returned in September 2004 and flew by just 1.5 million kilometers (930,000 miles) from Earth's surface.
In March 1989, a 300 meter (975 foot) asteroid crossed the terrestrial orbit and missed the Earth by just six
hours. Astronomers spotted the rock only when it was receding into space.
An asteroid measuring over 1,000 meters in diameter is potentially capable of destroying human civilization.
Chances of a major asteroid impact in the 21st century are a mere 0.0002 percent, although there is a 2 percent
probability of Earth colliding with a 100 meter asteroid before the year 2100.
The blast would equal to 100 Megatons in trinitrotoluol equivalent, and it would kill millions of people if it hit a
populous industrial region harboring many hazardous enterprises.
Recent close approaches of comets and asteroids
Comet Lexell, discovered in 1770, was the closest approach of any comet so far. The comet was
perturbed by large gravitational field of Jupiter, brought in close to the Earth (0.1 AU)
and then thrown out by Jupiter into a long orbit never to be seen again.
Adonis,
discovered in 1936, passed within 0.015 of the Earth, lost, and rediscovered in 1977.
Hermes,
discovered in 1937, came within 0.006 AU, never to be seen again.
Icarus,
discovered in 1949, passed within 0.04 AU in 1968. Every nineteen years the large
asteroid Icarus swings by planet Earth, often coming within four million miles of the
planet—mere spitting distance in astronomical terms. Icarus last passed by Earth in
1997. Before that, its previous approach was in June 1968. We now know that such
near-Earth asteroids are not all that rare and in recent years Congress and NASA have
shown greater interest in trying to track, and even visit them.
Apollo, discovered in 1932, when it came within 0.07 AU of the Earth, lost and rediscovered in 1973
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
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Fig. 7 Comet Lexell
Fig. 8 Asteroid Hermes
The most recent discovery: Asteroid 1991 BA, came within 0.0052 AU of the Earth when first seen, then its
approach decreased to 0.0033 and as it passed the Earth-Moon system, it was only 0.0011 AU away.
The best way to get a sense of the distances involved is to convert these distances to lengths equal to the
distance between the Earth and the Moon, 3.86x108 m, or 386, 000 km. Which one(s) of these close approaches
are within 1 Earth - Moon distance?
Evidence of past impacts are clear and the presence of near-Earth asteroids show future impacts are
possible, but the absolute probabilities are low. Nevertheless, the consequences of an impact could spell the end
of civilization. For example, a repeat of a small Tunguska-type impact over a populated area would kill almost
100,000 people. On the basis that the risk of dying from an impact is roughly the same as dying in a plane crash
a number of investigators suggest that the government (US) donate about $125 million a year toward the study
and assessment of the asteroid hazard. The present allotment toward this goal is less than $2 million. The
research in this area should be, of course, an international effort, since everyone on the globe is equally involved.
IL ** Move an asteroid 2008
http://www.spacegeneration.org/asteroid
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
ASTEROID AND COMETS IN THE MEDIA
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In the last twenty years there has been an explosion of activity, observational reportage and media coverage of
the dangers of asteroids or “interplanetary fugitives”. Although public awareness has been created of the high
probability of a direct collision, most of the public does not seem to have the necessary scientific literacy to
understand or assess this danger.
Well-made science fiction films benefit the general public in two ways: they can educate, increasing the
scientific literacy and they can also promote a positive emotional response to support scientific and technological
problems of common and global need and interest. Public concern and interest that is based on sound scientific
information is able to have a great effect on funding of important research projects. The perceived awareness of
the probability and the ultimate inevitability of a major collision with an asteroid or comet has already convinced
the American government to finance groups of scientists such as the astronomers who are involved with Project
Spaceguard. Our discussion below is partly based on the fine article “Sci-Fi in the Classroom”, published in
Mercury, Dec., 1998, by Leroy Dubeck, a physicist and Rose Tatlow, a science educator, The following are two
appropriate quotes for us, one is taken from the beginning and the other is the concluding sentence in their fine
article:
The depiction of science - whether credible or incredible - coupled with an explanation of the relevant
scientific principles, help make abstract ideas concrete.
Science fiction films can do more than any lecture or book to garner interest in and support of the
sciences - both from the public and from students in our classrooms. Good luck in making a deep impact
with your audience!
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
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Fig. 9 In this scene from Deep Impact, people watch an asteroid zooming across the
sky. It will strike the ocean and generate enormous waves.
“Deep Impact”: The movie
The movie Deep Impact was released 1998. Steven Spielberg was co- producer, who is actually one of the
directors of the Planetary Society. It should be noted that in order to “get the science right” the producers hired a
number of consultants from space science and engineering”, including the late Gene Shoemaker his wife Carolyn
and a NASA astronaut. It will be assumed that you have seen the movie at least once, so that we need not go into
telling of the story and the describing the action in great detail. However, you should see Deep Impact again,
soon after our discussion. Hopefully you will then see this movie with “different eyes”.
At the start of the movie we meet a group of high school astronomy students are studying the sky and
one of them sees an unusual ”star” and immediately sends a photograph of the object to an astronomer for
identification. We soon learn that the “star” is a comet, seven miles (about 11 km) in diameter and on a collision
course with Earth. The comet is supposed to be “the size of Mount Everest”, certainly large enough to destroy
life as we know it. We are not informed of this impending disaster, instead we are given the code name E.L.E.,
for “Extinction Level Event”. This is probably done as a dramatic device of “foreshadowing”
We are told that the government has decided to try three plans to stop or to lessen the destructive power
of the comet. The first one is called “the Messiah Project” and it would try to send a spacecraft to the comet and
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
plant nuclear “mole” bombs within the surface of the comet in order to either destroy it or alter its course
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significantly. We are told that the spacecraft used for this important mission is propelled by the Orion propulsion
system, a system originally designed for interstellar travel. This system uses a series of small controlled nuclear
explosions to push against a massive metallic rear plate on the spaceship. There is, of course, a shock-absorbing
system designed to reduce the impact of the explosions on the crew members. The push on the plate adds large
increments of change in forward velocity. Unfortunately, we are not sure how these explosions are supposed to
occur, one large one , several large ones, or very many small ones. The astronauts are seen moving about the
ship in a weightless state and later they move about the comet surface the same way.
Once the astronauts are on the comet they try to plant the “mole” bombs in a race against time. There
seems to be very little gravity and as soon as Sunlight strikes the surface, powerful jets of gases are emitted.
They also find house-sized chunks of rocks surrounding the comet. The Messiah astronauts succeed in burying
five nuclear explosives equivalent to megatons of TNT 100 meters below the surface. However, the explosions
do not deflect the comet but manage to break the comet into two parts, both still on a collision course with the
Earth! The first part is about 2.5 km across while the second about 8-9 km.
The second plan calls for the destruction of both comets by firing ICBM’s at them as they near Earth, but
it also also fails. Unfortunately, we are not told or given any details as to why the mission failed or how many
nuclear missiles were fired . It is difficult to comment here on the realism of the film.
The third plan was to build vast underground “arks” in various countries in the hope that those who are
admitted to these places will survive long enough to emerge into a world that again is able to provide food and
shelter. We are not shown the interior of these arks, but told that few people in the ark (except for some doctors,
scientists, teachers, and artists) are over the age of fifty.
While all this is transpiring on Earth, the Messiah astronauts decide to embark on a suicide mission. As
one astronaut says: “Look at the bright side; we’ll all have high schools named after us”. Their plan is to destroy
the larger of the two newly formed comets by guiding the spaceship into a deep fissure and detonating a second
set of four hydrogen bombs. Fortunately for the human race they are successful.
Other recent and major collision (doomsday) films
The movie Meteor, starring Sean Connery, treats the same theme: a five-mile (8 km) rock is on a collision course
with Earth after an impact between an asteroid and a comet in the asteroid belt between Mars and Jupiter.
The human race is scrambling to find a way to stop the asteroid. The asteroid is finally destroyed by the
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co-operative efforts of USSR and the US. The nuclear rockets that destroy the asteroid are fired from both the
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American and Soviet missile platforms. These platforms were placed in space-orbits with weapon systems that
were originally directed against each other’s countries. The destruction of the asterois, however, is not complete
and splinters of the rock strike the northeastern coast of the United States, producing tidal waves that cause
destruction not unlike that in “Deep Impact”. New York City did not fare well in either Deep Impact or in
Meteor.
Another film which made an “impact” on the general public is Armageddon. The general opinion among
scientifically literate people is that this film was a much less scientifically accurate than the other two. This is
surprising because NASA allowed filming on one of its sites; the scene with the astronauts simulating “zerogravity” (actually free fall) was filmed in a real training tank. The plot, however, was unrealistic and implausible.
The government is informed about an asteroid “the size of Texas” being on a collision course with Earth.
What is implausible here is that this sighting was made only a week before the predicted collision! Clearly, most
people would realize that a thousand mile wide asteroid would have been discovered a long time ago.
As you know, Ceres was the first to be discovered (1801) and it is still the largest asteroid we know and
is only about 600 miles (1000km) across. Of course, a collision between such a large asteroid and the Earth
would clearly spell the end of life on Earth and there would be no defence against such catastrophic event.
The film had even a greater box office appeal then “Deep Impact” which is attributable to the action and
suspense created by the director as well as the universal popularity of the main actor, Bruce Willis. It is evident
that despite the implausible scenario presented in the movie, the theme of an impending global disaster and how
we would cope with such knowledge captures the imagination of the public.
Finally, there is a “Doomsday” episode in Star Trek: The Next Generation called In a Matter of Time
(Episode 209, original airing date 11/91). In this episode the Star Trek crew tries to cope with the disaster created
by a collision of a planet with an asteroid (in a different solar system, of course). A massive cloud dust is
produced by the collision causing a cooling effect, quickly starting an ice age on the planet. The crew then
attempts to create a “greenhouse effect” to counter the cooling effect of the dust.
Questions
1.
Is it realistic (plausible) that a student using a simple 4 inch telescope would be the first on
Earth to discover a comet of the size depicted in Deep Impact? In order to answer the
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question above you should answer the following
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2.
a.
What percentage of the sky is being continually scanned by professional astronomers?
b.
What nations are devoting resources to systematic searches for comets and asteroids ?
c.
How much funding is provided for such research in the US? In Cananda?
d.
Approximately how many potentially dangerous comets / asteroids are there?
Do you think that a potentially catastrophic event, involving a comet of the size of Halley’s
comet, could be kept secret in our information age for so long? Discuss.
3
How realistic is the depiction of living in the weightless state of the spacecraft? On the comet?
How
4.
were these effects achieved?
While the astronauts are drilling into the comet’s surface we see that as soon as the radiation of
the Sun strikes the surface powerful jets of gas are emitted from within the surface. Is this what
you would expect, given the composition of comets and the conditions of deep space?
5.
In the movie we see house-size rocks on the surface of the comet. Is there any evidence for
thinking that there are rocks of this size on comets? Discuss.
6.
We have already found that the albedo of a comet is typically about .03, or 3%. That means that
without the comet’s tail reflecting the Sunlight, we would have never discovered any comet.
How then was it possible for the audience to see the scenes when the astronauts were drilling?
7.
We are not shown the construction or the interiors of the “arks”. However, you can speculate
about the construction of such a habitat and the supplies necessary for surviving for a long time.
10.
How realistic is the description of the effect of the tidal wave that would result if a comet of the
size (and at relative speed of about 50 km/s) landed in the Atlantic ocean? Would only New
York and the immediate environment suffer? Discuss.
11.
A recent (1999) Internet report on astronaut commented on the credibility of science in movies:
..There was a sensational movie 1983 on a man -made asteroid trajectory accidentally put on a collision
course with Earth, but the Hollywood scenario was quite unrealistic. We can expect more of this, from
Hollywood in the near future, which will influence the perspective laymen not scientifically inclined for a
few sensational thrills and the monetary benefit of Hollywood elites. Nonetheless, one would expect that
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a quick response a rendezvous team would be set up to protect Earth in the long run against both man-
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made objects and naturally occurring asterois and big rocks that pass the Earth.
What movie is this statement referring to? Discuss this statement.
Problems
1.
We have already calculated the approximate gravity (m/s2 ) on asterois and planets. What
would be the gravity on a comet of the size and mass in Deep Impact, expressed as percentage
of the Earth’s surface gravity?
2.
The “Orion propulsion” system described in the movie uses a series of “small” nuclear
explosions to impart incremental changes of velocity to the spaceship. In Chapter 2 we
discussed the simple case of chemical explosions that were incremental, hurling equal masses
away from the rocket at a given ejection velocity. Compare that analysis of rocket propulsion
to the Orion propulsion system. Assuming that the mass of the spacecraft is a modest 10,000
tons and that you are in “gravity free” space, answer the following questions:
a.
What incremental mass ejection is necessary to increase the velocity by 10
m/s? Assume that the ejection velocity (relative to the spacecraft) is 1000 m/s.
b.
How many of these ejections would you need in order to change the velocity of the
spacecraft by 1000 m/s?
c.
Imagine one large nuclear explosion with the equivalent of 1000 tons of mass
ejection in a time of 1/100 of a second. What acceleration would that produce?
d.
Speculate on the probability of survival of the crew if an atomic detonation of that
equivalent magnitude were to take place.
3.
In the movie, the president of the US (played by Morgan Freeman) says, during a news
conference,
“ This comet carries the equivalent of 500,000 mega tons of TNT.” Knowing the dimensions of the
comet and assuming its density to be about 0.1 g/cm3 , comment on how realistic that number is.
4.
We discussed the rotating space station of the type used in the famous film 2001: A space Odyssey”
(1968) in Chapter II. Describe the “gravity” as an astronaut would feel it, moving from the center to
the rim along one of the “spokes”.
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5.
The smaller comet does hit the Earth, just off the Atlantic coast of North America, creating a tidal
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wave, or tsunami. This wave is first about 30 meters high and moves with a velocity of 1100 miles
per hour (or 1800 km per hour). On impact with the shore we see a tidal wave thousands of feet high
(1000 meters?) that sweeps over New York City and moves inland to about 600 miles
(1000 km). Do you think the “scale of things” makes sense, in view of our previous discussions
involving doomsday scenarios?
6.
For the movie Apollo 13 (1995) the producers were actually allowed to use NASA’s anti-gravity
simulator, the aircraft KC-135, to film the sequence in which the actors seem to be floating about the
cabin. The aircraft first reaches an altitude of 30,000 feet (use 10,000 m as an approximation), with a
speed of near Mach 1 (use 300 m/s as an approximation). The aircraft then descends, following
roughly a parabolic curve, ascends again and is able to complete many cycles, as shown in Fig. 33.
The people inside are actually in free fall for 23 seconds for each descending part of the cycle.
a.
At the very top the people feel weightless and then remain weightless for 23 seconds
b.
How far does the aircraft “fall” in a vertical direction the aircraft travel in the
horizontal direction during this time?
c
What is the radius of the arc of the circle necessary to ensure free fall?
d.
Assume that free fall is maintained for 23 seconds. Describe the trajectory for this
portion of the run.
e.
If the bottom part of the trajectory has the same arc as the top, what will the weight of
the astronauts be?
f.
If the run is symmetrical what is the total vertical height through which the aircraft must
manoeuvre during this time?
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
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Fig. 10 The NASA aircraft KC-135 in a trajectory to simulate weightlessness.
http://images.amazon.com/images/P/0124467601.01.LZZZZZZZ.jpg
IMPACT CRATERS ON EARTH
Only about 150 impact craters have been recognized on Earth. Not until about 1930 was the impact theory for
craters, such as the famous Barringer crater in the Arizona desert, accepted by geologists. It may come as a
surprise to you that the origin of lunar craters was also a matter of dispute until about the 1950s. The astronomer
Ralph Baldwin pointed out in 1949 that, because the Earth and the Moon were companions, the evidence of
thousands of impacts on the Moon points to a similar impacting on Earth. In some sense then the Moon presents
us with a better record of terrestrial impact cratering than the Earth does. Baldwin went on to describe the most
famous impact scar on the Moon, the Tycho crater (named after the sixteenth century astronomer Tycho Brahe) :
“The explosion that caused the crater Tycho would, anywhere on Earth, be a horrifying thing, almost
inconceivable in its monstrosity”. The Australian astronomer Duncan Steel suggests that “With this compelling
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warning, the modern era of catastrophism began in earnest”.
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By 1949 only half a dozen Earth-crossing asteroids were discovered, but astronomers now (1999) have
identified about 160 such asteroids. Of these half are more than 1 km across. It is estimated that more than 2000
Earth-crossing asteroids exist that are larger than 1 km across. Examination of lunar rock samples from the
Apollo mission (1970) finally convinced astronomers that the origin of lunar craters is not volcanic action but
high velocity impact with comets and asteroids. In addition, there was good evidence for believing that there is a
continuous bombardment by particles ranging from dust to basketball size. The Earth must undergo a similar
bombardment from objects in space. A simple calculation (see Questions below) shows that the number of
impacts per unit area on Earth should be about twice that we calculate for the Moon, because the larger gravity
on Earth should pull in more. Moreover, the total number of impacts on Earth (because of its larger surface area)
should be about 25 times as large as on the Moon. But where are these craters?
Clearly, the atmosphere protects us and many of the impactors burn up on the way down; the area of the
ocean is much larger than the land area, and geological erosion and sedimentation obliterates most traces of
impacts. The Moon, on the other hand, has no atmosphere or water (only traces, see Chapter VI) and erosion
does not exist. The footprints of the astronauts are still on the Moon, and will be noticeable millions of years
from now, barring, of course, a collision with a comet or asteroid where the footprints are.
Impact craters on Earth range in age from a few thousand to about 2 billion years. Impact craters are formed
when a meteoroid or an asteroid collides with the surface of the Earth. A body whose mass is more than about
1000 metric tons and travels at typical speed of 20 km per second (relative to the Earth), would go through the
atmosphere practically unhindered. A body less than about 100 tons, on the other hand, would decelerate through
the atmosphere to about 50% of its original speed. The impact pressure would be enormous, about 100
gigapascals (one million times atmospheric pressure), and the atmosphere in the vicinity of the bolide would
reach temperatures of several thousand degrees Celsius.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
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Fig. 11 Crater formation
Differences in formation of a simple crater and a complex crater. The central peak of the complex crater is
formed as a result of uplift of material stratigraphically beneath the crater, which rebounds in response to
compression caused by the impact. From Melosh (1989)
They are characterised by a simple bowl shape similar to that of the transient crater suggesting minor
gravitational collapse following impact. Simple craters generally have depth / diameter ratios of
between 1/5 (0.2) and 1/3 (0.33). One example of a simple crater is the Barringer Crater, Arizona
(Figure 2), which is 1.186km in diameter today.
Much of the material ejected from the crater is deposited in the area surrounding the crater. Close to the
crater, the ejecta typically forms a thick, continuous layer. At larger distances, the ejecta may occur as
discontinuous clumps of material. Some material that is ejected is large enough to create a new crater
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
when it comes back down. These new craters are termed secondary craters and frequently occur as lines
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of craters that point back to the original crater.
Material below the surface of the crater is significantly disrupted by the shock of the impact
event. Near the surface is a layer of breccia (a type of rock composed of coarse, angular fragments of
broken-up, older rocks). Rocks at deeper depths remain in place (and are termed bedrock) but are
highly fractured by the impact. The amount of fracturing decreases as the depth below the surface
increases. The energy of the impact typically causes some material to melt. In small craters, this impact
melt occurs as small blobs of material within the breccia layer. In larger craters, the impact melt may
occur as sheets of material.
Questions
1.
Discuss briefly the difference between simple and complex craters. Find the diameter to depth
ratio and explain how gravity influences the formation of craters. For example, on the Earth,
the transitional diameter is 2 to 4 kilometers, whereas on the Moon it is about 15 - 20 km.
2.
The Internet has a lot of information on craters and pictures of most of the craters on Earth. Can
you clearly categorize them as either simple or complex? Explain.
3.
Using sand in a box, throw metallic spheres at various angles and notice the shape of the crater
made. Discuss.
Problems:
1.
Estimate the percentage of impacts by asteroids and comets that occur in the oceans. How is it then
that the only underwater crater discovered is the 60 km wide and 50 million year old Montagnais
structure on the coast Nova Scotia?
2.
Discuss the following statement: The number of impacts per unit area occurring on Earth should be
about twice that on the Moon. Therefore the Earth should encounter about 25 times as many impacts
as the Moon.
3.
It is estimated that about 500 tons of space material fall on the Earth surface every day. Assume that
this rate has not changed in 3.5 billion years. How many tons of space debris has each square meter
accumulated? Comment.
4.
For decades, maps of South Australia have shown a 35 kilometer wide dry salt pan called Lake
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
Acraman. It was not until in 1986 that geologists realized that this is the residual basin surviving from
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a colossal impact about 600 million years ago. The true nature of the crater was not recognized for so
long because all tell tale signs have been worn away. Now geologists believe that the collision of the
crater also caused a shower of rocks that was ejected and rained down, more than 300 km away, in a
mountainous region called Flinders Range. At the time, that region was occupied by a shallow sea,
so that the ejected rock accumulated in the sediment that was laid down for eons, and eventually
buckled up to form the mountain range.
5.
Estimate the initial velocity of the ejected rocks and comment on and estimate the energy of the
impact.
6.
A simple analogue to the action involved when an impact crater is formed can be demonstrated by
dropping a sugar cube in a cup of coffee. As in a coffee cup, many craters show a central uplift, or a
spike. In fact, in large impacts, when huge pressured (mega and even gigapascals) and high
temperatures (over 1000 degree of Celcius) are involved, the rock behaves like a fluid, but is then
frozen into a characteristic rebound form. Sketch a picture of what you observe, or thought you
observed, in the coffee cup.
7.
Students can make models of craters in the classroom with a box, lined with a trash bag. Using flour
at a depth of about 10 cm, some dry powdered tempera paint and various sizes of marbles. Students
should look for crater features and then test the effect of different velocities, angles of impact and sizes
of marbles
Jupiter and the mystery comet Lexell
Until recently, astronomers believed that the Earth is safe from impacts because the large gravitational field of
Jupiter periodically “sweeps clean” the inner solar system. While it is true that Jupiter does “sweep clean” the
inner solar system, Jupiter is also instrumental in occasionally sending us comets and asteroids into Earthcrossing orbits
This attitude of false security is partly due to a well-known series of events that involved comet Lexell,
which was discovered in 1769. The famous French astronomer Charles Messier observed what seemed to be a
verylarge comet and found that two weeks later it made the closest approach by a comet ever observed, missing
the Earth by a mere 0.015 AU, or about two times the distance to the Moon. Messier was puzzled as to why a
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
comet that had a tail of millions of miles long was not observed before 1769. The Russian astronomer Anders
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Lexell, however, soon showed that the comet had a short-period of 5.6 years. He also suggested that the small
elliptical orbit had been produced by the close encounter with the high gravity of Jupiter. The comet made no
reappearance in 1772, as was expected, and the Paris Academy of Sciences offered a prize in for the complete
solution of this puzzle.
The puzzle was solved by the astronomer Burckhardt, aided by the brilliant young mathematician Pierre
Laplace. Tracing back the comet’s motion , Laplace found that it approached within 0.1 AU of Jupiter in 1767
and concluded that the high gravitational attraction threw the comet into a new orbit. He also showed that prior
to this encounter, the comet had a period of about 52 years and was therefore too far to be seen earlier.
The new orbital period of the comet was found to be 5.6 years which happened to be almost exactly half
that of Jupiter’s orbital period. In 1776, 5.6 years later, the comet should have arrived back to perihelion, but the
Sun was in the line of sight . In 1779, 5.5 years again, or about 11. 2 years later, both Jupiter and Lexell
met almost in the same relative place as they did in 1767. However, Jupiter was now slightly ahead of Lexell
pulled the comet into a longer orbit of about 20 years, never to be seen again.
So the mystery of the comet Lexell was solved, and in the words of the astronomer David Milne in a an
article, written in 1828 (see References):”...the discovery may certainly be looked upon as having brought to
light one of the most astonishing facts in the whole history of astronomy.”
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Fig. 12 The mystery comet Lexell
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Fig13. The story of Comet Lexell.
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
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Fig. 14 Comet Lexell sighted in 1769, commemorated by this contemporary etching
1.
Draw your own sketch to show and explain the close encounter of Lexell with Jupiter and
describe how the trajectory of the comet changed twice.
2.
Explain to a friend how Jupiter redirected the motion of the comet, once toward the Earth, and
then away from the Earth.
3.
The unexpected motions of comet Lexell were first brought to the attention of astronomers in
general by an
article written in 1828 by the astronomer David Milne. In this article he says:
“At a time of their nearest approach, in August , Jupiter was distant from the Comet only 1/491
of its distance from the Sun, and hence exerted upon it a force of 225 greater .” Check this
claim and try to confirm it or reject it.
4.
Based on the story of the comet Lexell, discuss the role Jupiter plays in “controlling” the comet
and asteroid distribution in the solar system.
5.
After you have studied the section on “gravitational slingshot” and “gravitational breaking” on
page you will be able explain how the gravitational pull of Jupiter was able to change the orbit
of Lexell; once to pull it toward the Earth, and then to sweep it out into a very eccentric orbit,
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
never to be seen again.
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Fig. 15. Explanation of gravity slingshot effect
1.
The celestial body is “infinitely” more massive than the object approaching it. Therefore, the celestial
body’s frame of reference can be considered an inertial frame. We consider the Earth as a good inertial
frame (in spite of the fact that it rotates!). The celestial body can be considered to be moving at a
constant velocity (constant speed in a straight line), even though it may be orbiting a larger body (the sun
, in the case of the Earth, and the Moon, in the case of the Earth), as long as the time interval considered
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LCP 11 Part II: ASTEROID / EARTH COLLLISIONS
is small.
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2
. Energy and angular momentum are conserved in a closed system. That means that when an object falls
toward a large celestial body from very far away (but not influenced by an other large celestial body)
with an initial speed of V will leave the large celestial body with the same speed V as measured at a
very large distance from it.
3.
If a an object approaches the large celestial body too rapidly, deflection decreases and if the object
approaches too slowly, it will tend to crash into the large celestial body.
4.
To increase its velocity the object must approach the large celestial body from behind; to decrease its
velocity it must be approached from the front, that is from the direction in which the large body is
travelling around the sun. the direction of about 45 degrees to the orbit, to 100-1000 x the original
magnitude of that force, and 90 degrees to the motion, depending on where we choose place the orbit
(See Fig. 29).
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As before, we can find the approximate velocity of the object at the start of the Earth’s “sphere of
influence” as we have defined that region.. From here the trip will only take about 8 hours to reach the closest
approach to Earth.
36
Fig. 16 Gravitational assistance by the Moon to place the asteroid into the orbit of the Earth
Fig shows a typical lunar assist. The object comes in and catches up with the Moon, and swings around it,
reducing the speed (relative to the Earth only!) by about 1 km/s. The object ( a payload form Eros in our case)
arrives in the vicinity of the Moon, say at the height of 1 lunar radius (1.73x107.m). Using the same technique as
described in Fig. above.
Playing “orbital billiards: Capturing of the payload by using several Moon flybys.
In the game of “orbital billiards” we are tapping the gravitational energy source as our payload exchange orbital
momentum with the Earth and the Moon: the payload slows down while the Moon “speeds up”. The effect on
the Moon, as we have already mentioned, is very tiny indeed and cannot be noticed.
It is clear now that if we want to capture into a conveniently accessible orbit around the Earth a payload
from Eros, we will have to use a double or even a triple “lunar assist”. We will look at only the first stage of this
problem. One of the reasons for not going in to detail is the fact, that when discussing the lunar assist
trajectories in detail we are dealing with a three-body problem. This is a notoriously difficult problem , as we
have already indicated when discussing the libration points of Jupiter an the Moon. Essentially, Kepler’s laws
and the vis viva equation do not apply anymore and we would need to use complex numerical methods of the
type we applied when we discussed the calculations we used in determining the dynamics of bolides interacting
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with the atmosphere.
37
Doomsday reports
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38
Fig. 17 An example of a “doomsday” prediction.
A example of recent ‘dooms day' reports:
IL ****
http://www.washingtonpost.com/wp-dyn/articles/A38306-2005Apr8.html
Astronomer David Tholen spotted it last year in the early evening of June 19, using the University of Arizona's
Bok telescope. It was a new "near-Earth object," a fugitive asteroid wandering through space to pass close to
Earth.
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Tholen's team took three pictures that night and three the next night, but storm clouds and the moon blocked
further observations. They reported their fixes to the Minor Planet Center in Cambridge, Mass., and moved on.
39
They had never measured anything as potentially dangerous
to Earth. Impact would come on Friday the 13th in
April 2029.
The holidays and the tsunami in South Asia pushed 2004 MN4 out of the news, and in the meantime additional
observations showed that the asteroid would miss, but only by 15,000 to 25,000 miles -- about one-tenth the
distance to the moon. Asteroid 2004 MN4 was no false alarm. Instead, it has provided the world with the best
evidence yet that a catastrophic encounter with a rogue visitor from space is not only possible but probably
inevitable.
It also demonstrated the tenacity of the small band of professionals and amateurs who track potential impact
asteroids, and highlighted the shortcomings of an international system that pays scant attention to their work.
Doomsday calculations
The great detail and the data of the extensive research of the Barringer crater impact, Tunguska event and the
Yucatan impact give us good background knowledge for the calculation of damage that could result from
various sizes and kinds of impacts. We will look at these explosions and make some relatively simple
calculations.
1.
Studies of the damage caused by nuclear weapons shows that an airburst causes more
destruction than a ground burst and that the area of the ground laid to waste varies as the
energy of explosion raised to the power of 2/3. This is a purely empirical formula
(based on observation and measurement), and is not connected to a clearly articulated
theoretical background. Scientists often use such simple formulas as afirst “attack” in trying to
understand a problem According to the astronomer Chapman, the Tunguska event, which
occurred at an altitude of about 8 km, provides a good calibration for such events:
A = 400 Ex,
x = 2/3
where A is the area of devastation in square kilometers, and E is the energy of the explosion in
million tons of TNT equivalent.
Check this formula for the case of Tunguska. The area of devastation was about 2200 square kilometers.
Remember, the bolide is thought to have been about 50 m across and the kinetic energy was supposed to have
been somewhere between 10 and 20 Mt of TNT equivalent. Comment.
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LCP 11: ASTEROID / EARTH COLLISIONS
2.
Moving up in size, say to about 100m, consider Tunguska-like bolide, coming in at 22 km/s.
a.
Show that the energy40
of the impact is about 100 Megatons of TNT equivalent.
b.
Show that, according to our empirical formula, the area of devastation would
be about 10,000 km2.
c.
Pick a large city, such as New York, London, or Toronto. From a city map
estimate the area of the city you have chosen. How much of the city
would be destroyed if such an impact took place?
Doomsday scenarios
We will now look at seven doomsday scenarios. For each case, estimate the destruction involved, considering
a. The number of casualties based on the idea of “global average”,
b. The number of casualties based on the population density of a large city of your choice
40
41
The average population density of the Earth is about 3 per square kilometer, from Hong Kong (about
5000) down to about 2 in Canada and 1.5 in Australia. Because approximately about 70% of the Earth’s surface
is covered by water, the global average is about 10 people per square kilometer. Use our model, but you should
ask questions that the model does not answer and then try to apply your knowledge of physics to answer them.
Scenario 1:
A Barringer crater-like impact on the land. A 100 m bolide (iron-nickel) strikes the
Earth surface.
Scenario 2:
A Barringer crater-like impact in the ocean. !,000,000 people will die,
because of the effect of a tsunami on coastal population.
Scenario 3:
A Tunguska-like explosion: A 100 m bolide (stone) exploding in the
atmosphere over land is expected to kill about 100,000 people. Greater
land devastation than in case 1.
Scenario 4:
A Tunguska like explosion over the ocean. 1,000, 000 people will die because of
tsunami effects It is estimated that a tsunami is able to transport this energy at
very high speeds to coastlines, where the population density is usually high. A
small bolide impacting over the ocean then will probably cause more deaths tone
that falls over land. The estimate made by experts in the field is that in the first
case we can expect about 100,000 deaths and in the second about 1,000,000.
Scenario 5:
A 1 km asteroid (stony), enters the atmosphere at 15 km/s. and falls on land.
A 1 km asteroid collides with the Earth with a kinetic energy of about 0.1 to 1.0
million Mt of TNT equivalent (1012 tons of TNT equivalent).This is considered the threshold
energy for creating a “nuclear winter”, that is, enough dust (micron sized and smaller) is raised into the
stratosphere to block out the Sunlight so that the surface temperature drops by several degrees. This
amount of dust would take many months to settle out leading to a collapse of life in general. About
25%-50% of the human race would perish. Compare this to the potential destructive effect of the estimated
combined nuclear warheads in the world.
Scenario 6:
A 1 km asteroid (iron-nickel) falls into the ocean at 15 km’s and falls into the
ocean.
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Note: It turns out that this is about the smallest size of a space object we can
detect and tract with confidence. There are good reasons to believe that astronomers could
detect and tract all
Scenario 7:
of them in about 20 years.
A Yucatan impact-type falls on land: A 10-20 km asteroid collides with the
Earth at about 15 km/s.
Scenario 8:
A Yucatan impact-type falls in the ocean. A 10-20 km asteroid falls into the
ocean at about 15 km/s.
Table 7:
Impact
Scenario
Energy
Involved
(TNT equ.)
Altitude
of
Explosion
Environmental
Stresses
Casualties:
Heavily populated
--------------------------Global average
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Scenario 1.
100 m
asteroid
(15 km/s)
landing on
ground
-----------------------
Scenario 2.
100 m
asteroid
( 15 km/s)
landing in ocean
----------------------
Scenario 3.
100 m
asteroid
(15 km/s)
exploding in
atmosphere
----------------------
Scenario 4
100 m
asteroid
(15 km/s)
exploding over
ocean
-----------------------
Scenario 5
1 km
asteroid
(15 km/s)
falling on land
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Scenario 6
1 km
asteroid
(15 km/s)
lands in ocean
Scenario 7.
---------------------
10 km
asteroid
(15 km/s)
lands on the
ground
Scenario 8.
10 km
asteroid falling
in ocean
------------------------
Note: We have only considered asteroids in our scenarios. Most asteroids under about 100 m will explode high in
the atmosphere. You can check this using our model. However, you may want to consider what would happen
when a large asteroid like Halley’s comet would collide with the Earth. Remember that most asteroids orbit
clockwise, that is, opposite to the motion of the Earth (comet Lexell was a notable exception). The entry
velocities of comets, therefore, are much higher than those of asteroids
(about 30-35 km/s).
Use our model and find out what would happen if Halley’s comet collided with the Earth, say head-on.
Consider the size of comet Halley to be about 10 km, with a density of about 0.1 kg/m3 . You should be able to
estimate the velocity of impact from the data given in the first section.
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Fig. 18. Size and frequency of collisions with Earth
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Fig. 19 Travelling to Eros
Brief description of the NEAR trajectory:
I: Launch Phase
Feb. 27,1996: Launch took place at 20.43 UT (3:43 p.m. EST) from Cape Canaveral, using a Delta II-7925
expendable launch vehicle.
!The Delta II parking orbit (low Earth orbit (LEO) had an altitude of 183 km and an inclination of
28.74 degrees.
!The launch azimuth was fixed at 95 degrees.
!The coasting period in orbit was 13 minutes, allowing solar power to be used, one hour
after launch.
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!The injection burn, powered by the third stage solid motor, lasted for 4 minutes. The space craft (SC)
was entirely inside the Earth’s shadow.
!Approximately 22 minutes later the SC separated from the third stage.
!A yo-yo de-spin mechanism released the stowed position of 69 to 0 rpm and simultaneously released
the solar panels.
!After the third stage, responsibility for attitude control was shifted to the SC guidance and control
subsystems.
!The SC left the Earth’s shadow 37 minutes after launch. Up until now the space craft was battery
powered.
II: Cruise Phase
(Remember: Eros orbits the Sun at an angle of 10.8 degrees to the ecliptic).
!The spacecraft followed the “Delta VEGA” trajectory, short for “change in Velocity, Earth Gravity
Assist”. This was necessary to provide the extra energy needed to rendezvous with Eros.
!During the flight, low-level burns were performed to calibrate the propulsion system and to correct for
any trajectory errors.
!The space craft was put into “hibernation mode”.
June/July 1997 period: NEAR maintained this course until preparations were made for the two
upcoming critical events - the Mathilde flyby and the Deep Space Manoeuver, when encountering Eros.
Feb. 18, 1997: NEAR established record for the greatest distance from the Sun for a solar-powereed
spacecraft at 327 million kilometers, or 2.18 AU.
June 27, 1997: The spacecraft flew within 1200 km of Mathilde at 12.56 UT with a relative velocity of
9.93 km/s, returning images and other data.
July 3. 1997: NEAR decreased the velocity in a two-part burn of the main 450 N
thruster.
This action decreased the velocity by 279 m/s and lowered the perihelion from 0.99 AU to 0.95 AU.
January 23. 1998: Earth gravity assist swing-by took place at 7:23 UT (Universal Time) and placed the
SC on its final approach with asteroid 433 Eros. Closest approach to Earth was about 500 km, above
Ahvaz in South-Western Iran. This manoeuver changed the orbital inclination from 0.5 to 10.2 degrees
and also changed the aphelion distance from 2.17 to 1.77 AU in order to match those of Eros.
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April 1, 1998: NEAR sets the record for being the most distant manmade object detected by optical
means. An amateur astronomer in Australia spotted the SC at a distance of 33. 65 million kilometers (
.22 AU). The previous record was the 1992 sighting of the Galileo spacecraft at a distance of 8. 0 million
kilometers (.05 AU) from Earth.
Dec. 20, 1998: Attempted a 15 minute bi-propellant engine burn to align the SC with the position and
velocity of Eros. However, burn had to be aborted and spacecraft entered Earth Safe Mode (ESM) orbit
About an hour later, a low voltage stage was detected and the SC was entered into the Sun Safe mode
(SSM) orbit.
Dec 22, 1998: The SC was first detected while it was in the SSM; it was then recovered and guided int
Dec 22, 1998: The SC was first detected while it was in the SSM; it was then recovered and guided into
an ESM mode and finally put back into operation mode. The Multispectral Imager (MSI) was activated
in preparation for the Eros flyby.
Dec. 23, 1998: The NEAR Infrared Spectrometer (NIS) and Magnetometer were turned on in preparation
for the Eros flyby. The closest approach of the Eros flyby took place at 18:42 UTC.
III: Insertion phase
Jan. 3, 1999: Deep Space Manoeuver (DSM) was accomplished by a successful biprop engine burn that
put NEAR on target for the Feb. 2000 rendezvous with Eros.
April 21. 1999: NEAR’s orbit brings it within 1.5 AU of the Sun. Reorientation of the spacecraft was
necessary to reduce thermal stress on the solar panels.
Aug. 12. 1999: Trajectory Correction Maneuver (TCM) 19 will change the speed
(relative to the Sun) by about 21 m/s.
Oct. 10 1999: Re-start of instrument operations will begin.
Febr. 2, 2000: Orbit Insertion Maneuver (OIM) of about 10 m/s.
\Feb. 14, 2000: Orbit Insertion Maneuver (OIM) involving an approach speed of about 10 m/s will place
the NEAR spacecraft in orbit around Eros.
Originally it was planned to begin a series of four rendezvous manoeuvers with the main thruster to slow NEAR
down by 949 m/s in order to achieve a relative velocity of 5 m/s. It was then scheduled to fly by Eros on its
sunward side at a distance of about 500 km. Studying the trajectory of the orbit will give scientists an
opportunity to determine the mass to a 1% accuracy, allow the identification of several hundred surface
landmarks, and determine the asteroid’s spin vector more accurately. A search will also be conducted for
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possible satellites of Eros. This investigation will have the capability of locating any bodies larger than 5 m..
After a couple of days NEAR will be parked in a circular orbit of about 200 km, finally tightening the radius to
about 35 km.
Details about the NEAR spacecraft that will assist you in problem solving
Near spacecraft details:
On-orbit mass: 805 kg (including 318 kg propellent)
Power system: Four solar panels1800 Watts @ 1 AU
Shape: Octagonal prism shaped, 1.8m x1.2 m gallium arsenide solar panels
Fixed 1.5 m X-band high-gain radio antenna
Propulsion: A 450 N main thruster, a 20 N and a 5 N thrusters, total Delta V of 1450 m/s.
The system carries 209 kg of hydrozine and 109 kg of TNO axisizer in two oxidizer and three fuel tanks.
Space craft guidance is achieved through the use of a sensor suite consisting of five digital solar attitude
detectors, an inertial measurement unit AMU) that contains contains hemispherical resonator gyros as
well as accelerometers. There is also a star tracker camera, pointed opposite to the instrument that
establishes a direction.
Fig. 20 Science instruments used on flyby.
NEAR’s six science instruments:
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The magnetometer (MAG) will determine the strength of a magnetic field around Eros, if one exists.
A magnetic field would be strong evidence for the asteroid containing abundant metallic iron. Many of
the recovered meteorites contain iron.
The X-ray/gamma-ray spectrometer (XGRS) will determine the existence and the quantity of elements
such as iron, uranium, thorium, and potassium.
The near-infrared spectrometer (NIS), will analyze the reflected light and map the mineral
composition of Eros.
The laser rangefinder (NLR), actually a “laser radar”that is an altimeter, that will determine the shape
of the asteroid with an accuracy to a few meters.
The multispectral imager (MSI) will map the shape , landforms, and color properties of Eros in order to
determine the configuration of its different types of rocks and determine the processes that have shaped
the surface.A radio science experiment (RS) will track the tiny changes in NEAR’s radio frequency
caused by the changes in velocity (using the Doppler effect) as the spacecraft responds to the gravity of
Eros. This response will determine the mass of the asteroid. Establishing the volume, as determined by
imaging, will allow scientists to find the density of Eros.
Questions
You are expected to find the information about NEAR EARTH ASTEROID RENDEZVOUS (NEAR) on the
Internet. There is a lot of detailed information available. Read the information with the following guiding
questions in mind:
1.
Why was Eros chosen as the “ideal” asteroid to study? Give several reasons for
this choice.
2.
What are the specific goals of the NEAR mission?
3.
Astronomers in the early 1800 thought they were observing an atmosphere on the
large asteroids, especially on Ceres. Why do astronomers now believe that
neither Ceres nor even the much larger Moon could support an atmosphere?
4.
Using the Internet, find the NEAR education page and describe one of the
instruments in detail and discuss what its function is for the NEAR.
Problems
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1.
Estimate the mass of Eros, based on the data given above.
2.
Assume that Eros is a sphere and calculate the surface gravity. What percentage
of the Earth gravity is it?
3.
Calculate the escape velocity on Eros. Could you throw a ball into orbit?
4.
The closest approach of Eros to Earth in the twentieth century was on January 23,
1975, at about 0.15 AU.
5.
a.
How many ‘Earth radii’ is that?
b.
How many ‘Moon distances’ is it?
It is interesting to find out what the attraction of the Earth on the asteroid was at \
the distance of 0.1 AU and compare that to the attraction the Sun provided.
Show that the gravitational attraction of the Sun is still much larger, in fact about
10,000 times as large.
6.
The Moon’s distance from the Earth is about 3.84 x 10 8 m, or about 60 Earth
radii. Imagine an asteroid
Coming as close as 120 Earth radii, or two Earth-Moon distances. This would be
about 0.0051 AU, a very close approach, indeed. Calculate the ratio of the
Earth’s and the Sun’s attraction. We can disregard the Moon’s influence at this
distance. Why?
7.
You can now calculate the Earth’s sphere of influence, that is, the distance
between the Earth and the Sun where the attraction of the Earth on an asteroid or
comet is equally strong. At this distance an asteroid like Eros would feel equally
attracted to the Earth and the Sun.Compare this distance to the distance from the
Earth to the Moon.
8.
In the above problem we disregarded the effect of the Moon. To show that we
really are not allowed to do that, consider the hypothetical situation below. Here,
we have a complicated situation where four bodies are involved. A problem like
this can only be solved by numerical methods; no closed analytical solutions are
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possible.
a.
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In the position shown, what is the instantaneous force (stated as a vector)
acting on Eros?
b.
Find the instantaneous acceleration on Eros.
c.
Do Kepler’s laws apply here? If not, why not?
Orbit calculations for NEAR spacecraft mission: A guided problem discussion
(Note: For all these problems consult Fig. , The NEAR trajectory. In addition, it is recommended that you
accurately sketch the orbits to scale on a large sheet of paper (60x40 cm is a good size), and mark your progress
as you go along. Use two pins and a string to trace your ellipses) Also assume that the orbits of Eros, Mathilde,
and the Earth are coplanar (we know that, actually, Eros has an inclination of 10.8 degrees and Mathilde 6.7
degrees with the eclectic, or the plane of the Earth’s orbit).
We will describe four orbits for our problem: The orbit of: the spacecraft, of asteroid Mathilde, the
asteroid Eros, and, of course the orbit of the Earth. Use 1 AU = 1.5 x 1011 m.
We will assume that
a)
all orbits are moving in the plane of the ecliptic, and
b)
that the orbit of the Earth is circular.
The first assumption leads to considerable errors, because the inclination of Eros is about 10 degrees and of
Mathilde a little more than 6 degrees. The second assumption will lead to very small errors. However, for
purposes of getting a rough idea of the motion, this is a pretty good approach. Of course, you would not be hired
by the NEAR research group if your calculations were based on this elementary approach!
Part B: The Mathilde flyby, or “Waltzing with Mathilde’
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Fig. 21 The Mathilde flyby
The NEAR spacecraft was then placed into an orbit that intercepted the orbit of Mathilde on June 27, 1997.
According to the NEAR report, the distance to the Earth was about 2.2 AU at the time of the flyby, the distance
to the Sun about 2.0 AU and the flyby velocity (relative to the Mathilde) was estimated at about 10 km/s, in an
“off-tangent” direction. The closest approach was 1200 km.
Note: Most of the information you will need for the following problems are found in Fig. 28.
`
1.
From your sheet, showing the orbits of the SC and Mathilde:
a.
Estimate the distance from the SC to the Earth and to the Sun, expressed
in AUs. Do your values agree with the NEAR report? Comment.
b.
Show that the orbital velocity of Mathilde at this time is about 23.6 km/s
and of the SC about 18.1 km/s
c.
Estimate the angle between the two orbital velocities and work out the
relative velocity between the SC and Mathilde. How close is your value
to the one given in the NEAR report? Comment.
T
he SC was then placed into an orbit that intercepted the orbit of Mathilde on June 27, 1997. According to
the NEAR report, the distance to the Earth was about 2.2 AU at the time of the flyby, the distance to the sun
about 2.0 AU and the flyby velocity (relative to the Mathilde) was estimated at about 10 km/s, in an “off-
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tangent” direction. The closest approach. was 1200 km.
2.
From your sheet, showing the orbits of the SC and Mathilde:
a.
Estimate the distance from the SC to the Earth and to the sun, expressed
in AUs. Do your values agree with the NEAR report? Comment.
b.
Show that the orbital velocity of Mathilde at this time is about 23.6 km/s
and of the SC about 18.1 km/s
c.
Estimate the angle between the two orbital velocities and work out the
relative velocity between the SC and Mathilde. How close is your value
to the one given in the NEAR report? Comment.
3.
Astronomers calculate the mass of an asteroid by direct observation of size,
then estimating the volume and the density; or by noting the perturbation
(deviation from the “normal” orbit) of the orbit when approaching a large body
like that of the earth. Would the close approach to a large asteroid, like that
Mathilde, significantly change the orbit, so that it is measurable?
a.
Compare the force of the sun on Mathilde (expressed in N/ kg) with the
force of Mathilde, using an estimated mass for the asteroid to be 1x1017 kg. Comment.
b.
Estimate the gravity (m/s2) on the surface of the asteroid.
c.
What would be the approximate escape velocity from the surface of the
asteroid?
4.
About one week after the Mathilde flyby, a Deep Space Manoeuver (DSM) was
executed. This involved one of two major burns expected from the main thruster
of 450 N force The SC had to be slowed down by 279 m/s in order to lower the
perihelion distance of the trajectory from 0.99 AU to 0.95 AU.
a.
How long did it take to slow down the SC by 279 m/s? Estimate the mass
of the SC at this point.
b.
Show that a Delta-V of 279 m/s is going to change the perihelion
distance from .99 AU to .95 AU.
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5.
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Consider the orbit of Mathilde and calculate the velocity of the asteroid (relative
to the sun) at the time of the encounter with the spacecraft.
6.
Now calculate the velocity (relative to the sun) of the NEAR spacecraft as it
passes Mathilde. From Fig. 28 you should be able to find that the spacecraft was
in an orbit whose semimajor axis was about 1.60 AU.
7.
The Internet report of the Mathilde encounter claims that the fly-by speed of
NEAR was 9.92 km/s (velocity of spacecraft relative to the asteroid). The
encounter took place at a distance of 2.19 AU. Calculate the velocity (relative to
the sun) of the asteroid Mathilde and of the spacecraft. Does your calculation
agree with this value? Comment.
Part C: Back to Earth for a swing-by
We have already encountered a swingby when discussing the strange case of comet Lexell. Gravity assist,
variously known as swingby or slingshot effect, is used to change the direction and/or the inclination of a
spacecraft orbit, to increase or decrease the speed of the SC. When the SC gets close to the planet, it enters the
“the sphere of influence” of the planet’s gravity. This can be defined as the distance at which the planet’s
gravitational pull on the SC becomes significant (see problem below). The speed of the SC then will increase to a
maximum when it gets closest to the Earth. Clearly, the speed at this point must be greater than the escape
velocity from this point. In effect, there is an exchange of angular momentum with the Earth and the Earth will
slow down, or speed up, a very tiny bit. When the SC climbs out of the Earth’s gravitational field its speed is
decreased. The conservation of energy principle requires that the incoming speed and the outgoing speed (at a
point far away, beyond the “sphere of influence” of the Earth), relative to the planet, be equal. (To understand
better what happens when a spacecraft flies by a large celestial body, a planet or the Moon, turn to the
Supporting Text, or ST).
NEAR scientists used the swingby as an opportunity to test the performance and calibration of the SC’s
six instruments and to practice co-ordinated multi-instrument observations of the type that will be used on Eros.
The SC’s solar panels reflected the Sunlight to Earth so that it could be seen with the unaided eye. NEAR
approached Earth at about 6.7 km/s and reached its reached its highest speed at about 13 km/s at its closest
approach to Earth. It was then 478 km above South West Iran at 11:23 a.m. on January 23, 1998. NEAR then
left the Earth at about 6.7 km/s.
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On first thinking about gravity assisted flyby, you might ask: “How can this be a slingshot effect, if the speed
does not change?” The clue to the answer is given by realizing that are looking for speed change relative to the
Sun. See Fig. for a more detailed explanation.
Problems
After the engine failure, the NEAR spacesraft was allowed to return to earth, the earth-gravity assist swingby
occurring on January 23 1998. The closest approach to earth was 540 km and the velocity of the spacecraft
(relative to earth) was 13 km/s.
1.
The spacecraft was travelling at about 6.7 km/s when it entered the Earth’s
“sphere of influence” .We can define the “sphere of influence” as the distance
from the Earth where the gravitational attraction of the Earth begins to exceed
that of the Sun.
a.
Show the orbital velocity of the SC before entering the “sphere of
influence” of the Earth was about 36 km/s.
b.
You can easily show that this happens when the SC is approaching
Earth at a distance of 2.6x108 m. This is approximately 40 Earth radii
away, or about 2/3 the distance to the Moon.
c.
At a distance of about 4 Earth radii, what is the gravitational influence
of the Earth, compared to that of the Sun?
You should have found that the gravitational attraction of the Earth at a distance of about 4 Earth radii is
less than 1% of that of the Sun.. So, we can now assume that SC has passed from its helio-centric orbit to a
geocentric orbit at about that time.
a.
If the spacecraft passed overhead, approximately how long do you think
would it be visible?
b.
Show that the velocity of the earth (relative to the sun) is about 30 km/s.
c
Now calculate the velocity (relative to the sun) that the spacecraft
would have if the earth had no
influence on it. Show that this is about 36 km/s.
d.
The earth’ gravity assist then changed the velocity from 36 to about
49 km/s.
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e.
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What was the % increase in kinetic energy provided by the sling-shot
effect of the Earth?
2.
The closest approach of the NEAR spacecraft was 478 km. The ground track was from Europe to
Hawaii, changing the inclination from 0.5 degrees to 10.2 degrees, and reducing the aphelion distance
from 2.17 AU to 1.77 AU, to enter an orbit that matches the aphelion distance of the orbit of Eros. This
is our orbit V, described above.
a.
We can now try to confirm the observation that the highest velocity of NEAR
was about 12.8 km/s, as it passed the point of closest approach. What iwould be
the velocity of the SC if it falls from a large distance (a distance of 60 Earth radii can be
considered “infinity”) toward the Earth and then passes the Earth at a
close approach of about 500 km? Remember the SC already has a velocity of 6.7
km/s relative to the Earth.
b.
When the SC reaches the closest approach, it is moving with a speed of almost 13 km/s.
We know that the escape velocity from the Earth surface is about 11 km/s.
What would have happened if the spacecraft had a velocity of, say, 9 km/s at the
point of closest approach?
c.
If an Earth satellite were in a circular orbit at a height of 478 km (the height of
closest approach) , what would be its speed?
d.
What additional velocity (what we have called Delta V) would be needed to have this satellite
escape the earth?
e.
Using a globe and a small sphere, like a marble, try to show the near-Earth
trajectory and explain to a fellow student i. the sling shot effect, ii. how you think
the change in inclination was produced.
d.
Using the orbit parameters for the SC, the Earth and of Eros, estimate the
incoming velocity and the out-going velocity of the SC.
3.
Discuss the following claim: Voyager 2 toured the Jovian planets. The spacecraft was
launched on a standard Hohmann orbit (HOT) transfer. Had Jupiter not been there at the
time to give considerable boost to the spacecraft by the gravitational slingshot effect,
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what would have happened to Voger 2?
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More about swingby
1.
The celestial body is “infinitely” more massive than the object approaching it.
Therefore, the celestial body’s frame of reference can be considered an inertial
frame. We consider the Earth as a good inertial frame (in spite of the fact that it
rotates!). The celestial body can be considered to be moving at a constant
velocity (constant speed in a straight line), even though it may be orbiting a
larger body (the sun , in the case of the Earth, and the Moon, in the case of the
Earth), as long as the time interval considered is small.
2.
Energy and angular momentum are conserved in a closed system. That means
that when an object falls toward a large celestial body from very far away (but
not influenced by an other large celestial body) with an initial speed of V will
leave the large celestial body with the same speed V as measured at a very large
distance from it.
3.
If a an object approaches the large celestial body too rapidly, deflection decreases
and if the object approaches too slowly, it will tend to crash into the large
celestial body.
4.
To increase its velocity the object must approach the large celestial body from
behind; to decrease its velocity it must be approached from the front, that is from
the direction in which the large body is travelling around the sun.
Part D: Catching up with Eros
About 200 days before catching up with Eros, visual contact was made with the asteroid. From here it was
possible to navigate “optically”. In addition, initial shape and rotation determination of Eros were attempted.
Unfortunately, on December, 20, 1998, just 21 days from its scheduled rendezvous with Eros, NEAR failed to
complete a crucial engine burn, leaving scientists and engineers frustrated and scurrying to save the mission.
The burn was supposed to put the SC on track for an orbit insertion around Eros.. The SC defaulted into
a safe mode, and waited for instructions from the NEAR operations center. Actually, the SC by now had lost 30
kg of fuel. Luckily at 7:30 p.m. contact was made with NEAR, much to the relief of the NEAR group. They now
had to wait for the SC to make a preprogrammed 360 degree sweep, looking for a signal from Earth. The group
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had only a 10 minute window of opportunity to locate the signal and then upload crucial commands for the SC.
Later, Tom Coughlin, the project manager remarked: “ They were the longest 27 hours of my life”.
The NEAR group now had to work round the clock to find out what went wrong and examine
alternative options. Luckily, these were worked out before, in preparation for just such an emergency. But they
were unable to save the mission but were faced with a new challenge: get as much as you can from a flyby of
Eros.
New programs were hurriedly written that would direct the SC to take images of the asteroid and collect
valuable data as it flew pas at a high relative speed. The group managed to get enough data to give mass, shape
and composition estimates. There was no evidence of a Moon orbiting Eros, larger than 100 m.
On January 3, a 24-minute engine burn successfully increased the speed of the SC by 932 m/s in order to
“catch” the asteroid. This was a critical maneuver since it used up 57% of the fuel, in order to closely match the
speed of Eros. NEAR and Eros were now travelling in almost identical orbits around the Sun. NEAR was closer
to the Sun at perihelion, on the “inside track”, and will catch up with Eros, hopefully on Valentine’s Day
(February 14) 2000.
Questions
1.
According to the NEAR research group, the scientists and engineers controlling the
mission turned a misfortune to their benefit. They point out that the chance of a close
encounter with an the asteroid before entering an orbit has actually been a lucky break.
The encounter gave them a chance to establish the dimensions and the mass of Eros more
accurately and this knowledge will improve the orbit insertion manoeuver next year.
a.
The new size of Eros is 38x13x13km and the mass was found to be about
2.12x1016 kg. Find the density of the asteroid.
b.
The previous estimate of the size of Eros was 40x14x14 km. What is the %
difference between the two measurements?
c.
Do you think the gravity around the asteroid is significantly different for the two?
Discuss.
2. Discuss the following statement from the NEAR group:
Although 433 Eros does not represent a threat to the Earth at present, despite its occasional close
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approaches, this situation is slowly changing. Orbital calculations show that the perihelion distance is
slowly decreasing and that a future impact is quite probable. However, this impact will not happen for
another 100 million years. When it does, it will cause a crater of up to several hundred kilometers in
diameter.
You could refer to the section that discusses “doomsday scenarios” and call this the ultimate disaster that would
surely wipe out civilization and probably all of life on Earth.
Problems
1.
Where was the SC and were was Eros when first visual contact was made?
2.
Approximately how far was Eros from the SC when first visual contact was made?
3.
On December 20, 1998, just 21 days before the planned rendezvous, engine failure
occurred. Where were the two bodies and was the approximate distance between them?
4.
When the NEAR group tried to contact the SC how long did it take the signals to travel
from Earth to the SC?
5.
The 24 minute burn by the main engine delivering 450 N force produced a Delta V of
932 m/s. Given this information, what was the approximate mass of the SC at this time?
Mining on Eros
Even a brief search on the Internet reveals a buzz of planning and organization aimed at investors daring enough
to claim the potential riches that could result from space exploration, especially in asteroid mining. A typical
advertisement is the following: “A very small asteroid, such as 3554 Amun at 2 km in diameter, contains
material worth approximately US $20,000 billion!” The promise of great riches has attracted numerous
companies, and new private commercial efforts are now being mounted to visit asterois and the Moon for
mining purposes.
Before those great riches are harvested , however, a lot of preparatory work will have to be done. First,
Eros will have to be studied for one or two years after the SC is placed in a stable orbit. The mass, density and
composition will be studied and determined. We only have an idea what the surface of an asteroid would be like:
silicate dirt, mixed with nickel-iron granules and volatiles, or pure metal and pure powder?
Landing on an asteroid and then later launching materials from an asteroid will be much different from landing
or launching on the Earth or on the Moon. In addition, landing on an asteroid could serve as a training flight for
future planetary missions.
The low gravity on the surface of an asteroid is good because it will take very little energy to remove the
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ore but it will also provide new challenges. Staying attached to and moving skilfully around on an asteroid might
be done using harpoons, a net, or even magnets. A spinning asteroid would pose problems for placing the solarpowered processing equipment so that it faces the Sun for a long time. For example, how do you stop the rotation
of a large rock with a mass of millions of tons?
The actual mining operation on an asteroid, however, would be much simpler than mining on Earth, or
the Moon. For example, we do not need heavy mining and transport machinery and we don’t need complex
processing equipment as we would on the Moon in order to get valuable materials.
What methods should be used for mining on an asteroid? Would strip mining work? Or tunnelling into
the asteroid? One novel ideas is to place a canopy around a strip mining area to collect the ore which would be
propelled by a “dust kicker’ into the canopy and taking advantage of the “centrifugal” force produced by the
rotation of the asteroid. Another suggestion is to drill into carbonaceous asterois (25% of near-Earth asterois are
probably dormant comets), much the way we do on Earth when we drill for oil and natural gas.
Processing
Asteroid material is expected to be exceptionally high quality and not needing much processing. Basic ore
processing will yield for material and volatiles, usually stored as ices because of the extremely cold temperatures
in the shade, as well as selected minerals such as glasses and ceramics. At the input chute, the ore will be ground
up and sieved into different sizes as a first step in the basic ore processing system. Simple mechanical grinders,
using a rocky jaw arrangement for course crushing and a series of rollers for fine crushing , will be arranged in a
slowly rotating housing to provide centrifugal force-induced movement of the material. In addition, vibrating
screens will be used to sift the grains for directing them to the properly sized grinders. Strong magnetic fields,
generated by electromagnets that draw the electric current to produce these fields from solar powered batteries,
will be used to separate the nickel-iron metal granules from the silicate grains. An alternate method might be to
place the material into magnetic drums, so that the silicates and the weakly magnetic materila would deflect from
the drum and the magnetic granules and pebbles would be attracted to the drum until the “scrape-off’ point is
reached. Repeated cycling then, using a variable magnetic field,will yield high purity nickel and iron which then
can be put into large bags.
After the mechanical-magnetic grinding, a so-called “ impact grinder”, or “centrifugal grinder” could be
used. A very rapidly spinning wheel pushes the material along its spokes and hurls it against an impact block.
Any silicate impurities that are still attached to the free metal are then shattered off. Drum speeds can be used
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that produce a centrifugal acceleration that will flatten the metal granules by impact. The small silicate particles
can then be sieved out.
The non-magnetic material can then be placed into a solar oven where the volatiles are “cooked out ”
Since we are in near zero gravity (but not zero!) and windless space, very large oven mirrors can be constructed
from aluminum foil. The gas stream can then be piped to tanks that are located in the cold shadow. They can be
placed in series, so that the furthest one is the coldest, thus achieving differential condensation.
To store the volatiles, thin and relatively lightweight spherical tanks could be sent and used for storing.
To send materials back to Earth, tanks could be manufactured from the iron nickel metals mined on the asteroid,
after the metals have been melted down in a solar oven.
Unused material can be cast into bricks (again, using a solar oven) and then used to shield the habitat
from solar radiation. Of course, the waste material could all be bagged and then ejected from the asteroid.
Remember, even on a large asteroid like Eros, the escape velocity is only about 10 m/s.
Finally, after “consuming” the asteroid, the equipment could be moved to the next asteroid to begin a
new mining engagement. Rocket fuel for the delivery trip back to Earth could be produced by separating oxygen
and hydrogen, mostly using the water available as ice. It may be possible to have hydrogen chemically bond with
carbon to produce methane and use this a fuel.
The equipment should be sent to an asteroid in advance of the asteroid mining crew, on a slower and
more fuel-efficient trajectory. Once in place and all its vital systems functioning, the crew will be sent. Their first
task, of course, will be to set up a radiation-protected environment, followed by a secure commuting system that
protects the space miner from “escaping” the asteroid by just making a too large a jump by mistake.
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Fig. 22. Mining on a large asteroid
Questions
1.
Travelling, then living and working on an asteroid, and finally going back to Earth will take a
very long time. How long?
2.
Gravity on any point on the surface of a spherical asteroid would be easy to determine. But what
would the gravity on the surface of an asteroid that is shaped like a peanut? Discuss.
Problems:
2.
Would it be a practical proposition to try to stop a small asteroid rotating? To answer that
question, we must first answer the following: How much energy would it take to stop a small
asteroid from spinning? To calculate this energy, consider a cylindrical asteroid, a 1000 m long
with a diameter of 300 m and a density of 3 g/ cm3. The cylinder is rotating along its axis once
every 10 hours. The energy required to rotate a cylinder is given by the rotational analogue to
(linear) kinetic energy,
E = ½ I 2,
where I is the moment of inertia (the analogue to mass) and  (radians per second) the rotational speed.
The moment of inertia of a cylinder is ½ M r2, where M is the mass (kg) of the cylinder and r the radius.
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To get a sense of how much energy this is, convert into TNT equivalent. Comment.
Returning to Earth
After the mining has been accomplished, several very large bags of goods mined on Eros will be well packaged
and fitted with retroactive rockets. Next, we must escape the gravity of Eros and then slowly pull away from the
large rock, while staying roughly in the same orbit.
First attempt to find the point of trajectory change, with low Delta-V requirement:
The most direct reasoning would go like this. We could approach the Earth at a place where the orbital distance
separation is the smallest. It so happens that the orbital velocities here are almost equal, in other words, the
relative velocity is the lowest. Referring to your sketch of the orbit of Eros and the Earth, you can easily find
where the closest approach can occurs. When you have found this, you should determine the approximate
separation between the payload and the Earth. This distance will be at least .2 AU, which is about 3x1011 m. To
get a better sense of this distance, we can express it in terms of the distance to the Moon, 3.86x108 m, or about
780 times that distance. The payload must be redirected, using retroactive rockets, to fall into the Earth’s “sphere
of influence”. This still a long distance from where we start our transition orbit. It will take a long tome for the
payload to “fall into the Earth’s sphere of influence”. A quick calculation of the energy required to redirect the
object to go from Eros’ orbit to the Earth’s orbit ,when the Earth and Eros orbits are about .2 AU apart, is
equivalent to retroactive burning to produce a Delta -V of 17 km/s ! To accomplish a Delta-Vee of that
magnitude would require too much energy.
Second attempt: reconnect with the Earth’s orbit
You will remember that the first orbit of the NEAR spacecraft crossed the orbit of Eros. After a little reflection,
you will realize that at this crossing we could redirect the payload (and our spacecraft) and insert it into the first
orbit of the NEAR spacecraft. Using the vis-viva equation you can quickly find that, here, the relative velocity
between the two orbits is only about 1 km/s. An orbit insertion, going from the orbit of Eros to that of the first
orbit of the NEAR spacecraft.
This manouvre will put the payload and the our own SC , into an orbit that connects with tthat of the
Earth at perigee. As before , when we investigated the flyby of the NEAR spacecraft, we can decide how to
capture the payload as well as our own returning SC containing the “asteroid miners”.
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Capturing the payload, using an Earth flyby
As before, when we discussed the NEAR spacecraft flyby, the returning SC and the payload come closer to the
Earth, and enter the Earth’s “sphere of influence” at a distance of about 2.6x108 m. As we have seen before, this
is very close to the Earth, only about 2/3 the distance to the Moon.
From here on, the force acting on the payload and the SC will vary from 8.3 x10-3 N / kg, in the direction
of about 45 degrees to the orbit, to 100-1000 x the original magnitude of that force, and 90 degrees to the motion,
depending on where we choose place the orbit (See Fig. 29
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As before, we can find the approximate velocity of the object at the start of the Earth’s
“sphere of influence” as we have defined that region.. From here the trip will only take about 8
hours to reach the closest approach to Earth.
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Playing “orbital billiards: Capturing of the payload
by using several Moon flybys.
In the game of “orbital billiards” we are tapping the gravitational energy source as our payload
exchange orbital momentum with the Earth and the Moon: the payload slows down while the
Moon “speeds up”. The effect on the Moon, as we have already mentioned, is very tiny indeed
and cannot be noticed.
It is clear now that if we want to capture into a conveniently accessible orbit around the
Earth a payload from Eros, we will have to use a double or even a triple “lunar assist”. We will
look at only the first stage of this problem. One of the reasons for not going in to detail is the fact,
that when discussing the lunar assist trajectories in detail we are dealing with a three-body
problem. This is a notoriously difficult problem , as we have already indicated when discussing
the libration points of Jupiter an the Moon. Essentially, Kepler’s laws and the vis viva equation
do not apply anymore and we would need to use complex numerical methods of the type we
applied when we discussed the calculations we used in determining the dynamics of bolides
interacting with the atmosphere.
Fig 34 ? shows a typical lunar assist. The object comes in and catches up with the Moon,
and swings around it, reducing the speed (relative to the Earth only!) by about 1 km/s. The object
( a payload form Eros in our case) arrives in the vicinity of the Moon, say at the height of 1 lunar
radius (1.73x107.m). Using the same technique as above:
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Fig. 23 Using the Moon as a flyby.
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