Module 4: The Wanderers
Activity 2:
The Universal Force of Gravity
Summary:
In this Activity, we will investigate
(a) elliptical orbits and Kepler’s Laws,
(b) Newton’s Law of Gravitation, and
(c) apparent weightlessness in orbit.
(a) Elliptical Orbits and Kepler’s Laws
Some orbits in the Solar System cannot be
approximated at all well by circles
- for example, Pluto’s separation from the Sun
varies by about 50% during its orbit!
According to Kepler’s First Law, closed orbits around
a central object under gravity are ellipses.
As a planet moves in an elliptical orbit, the Sun is at one
focus (F or F’) of the ellipse.
v
r
F’
C
F
The line that connects the planet’s point of closest approach
to the Sun, the perihelion ...
v
perihelion
r
F’
C
F
… and its point of greatest separation from the Sun,
As a planet moves in an elliptical orbit, the Sun is at one
the aphelion
focus (F or F’) of the ellipse
v
perihelion
is called the major axis
of the ellipse.
r
F’
aphelion
C
F
The only other thing we need to know about ellipses is how
to identify the length of the “semi-major axis”, because that
determines the period of the orbit.
“Semi” means half, and so the
semi-major axis a is half the
length of the major axis:
v
r
F’
C
a
F
a
For circular orbits around one particular mass - e.g. the Sun we saw that the period of the orbit (the time for one complete
revolution) depended only on the radius r
- that was Kepler’s 3rd Law:
For objects orbiting a common
central body (e.g. the Sun)
in approximately circular orbits,
the orbital period squared is
proportional to the orbital radius
cubed.
M
r
m
v
Now here’s the mathematics….
We can write Kepler’s third law:
the orbital period squared is proportional
to the orbital radius cubed
in mathematical notation as:
radius
R
m
Period
P
where G and 4p2 are constants
and M is the mass of the Sun
This comes from equating the gravitational and centrifugal
forces:
=
and noting that the circular velocity:
Let’s see what determines the period for an elliptical orbit:
For elliptical orbits,
the period depends
not on r, but on the
semi-major axis
a instead.
v
r
F’
C
a
F
a
It turns out that Kepler’s 3rd Law applies to all elliptical
orbits, not just circles, if we replace “orbital radius”
by “semi-major axis”:
For objects orbiting a common
central body (e.g. the Sun)
the
the orbital
orbital period
period squared
squared is
is proportional
proportional to
to
the
radiusaxis
cubed.
the orbital
semi-major
cubed.
Each of these orbits
has the same
semi-major axis
length a:
Note that a circle
is a special case of
an ellipse, where
r = a.
So if each of these orbits is around the same massive
object (e.g. the Sun),
then as they all have the same
semi-major axis length a,
then, by Kepler’s
Third Law, they
have the same
orbital period.
Click here to see
a simulation illustrating
Kepler’s Third Law.
So, as you saw in the simulation, bodies orbiting at
large distances have much longer orbital periods.
For the mathematically
inclined, the square
of the period P of the
orbit increases in
proportion to the
P
cube of the
period
semi-major
axis a:
distant planets have
much large orbital periods
a semi-major axis
We haven’t yet met Kepler’s Second Law.
That’s because it’s not at all interesting for circular, or
almost circular orbits.
But if we look at a quite eccentric elliptical orbit,
for example, that of Halley’s comet:
Comet Halley in 1910
Neptune
Sun
Comet Halley
Note that Comet Halley’s orbit is retrograde, which means that
it orbits the Sun clockwise when viewed from the north pole.
This is the the opposite sense to that of the planets.
An object in a highly elliptical orbit travels very slowly
when it is far out in the Solar System,
… but speeds up as it passes the Sun.
According to Kepler’s Second Law,
… the line joining the object and the Sun ...
… sweeps out equal areas in equal intervals of time.
equal areas
That is, Kepler’s Second Law states that
the line joining a planet and the Sun sweeps out
equal areas in equal intervals of time.
(b) Newton’s Law of Gravitation
We call the force which keeps the Moon in its orbit
around the Earth gravity.
Sir Isaac Newton’s conceptual leap in understanding
of the effects of gravity largely involved his realisation
that the same force governs the motion of a falling object
on Earth - for example, an apple - and the motion of the
Moon in its orbit around the Earth.
Isaac Newton discovered that two bodies share a
gravitational attraction, where the force of attraction
depends on both their masses:
MSun
MEarth
Both bodies feel the same force, but in opposite
directions.
MSun
MEarth
This is worth thinking about - for example, drop a pen to
the floor. Newton’s laws say that the force with which
the pen is attracting the Earth is equal and opposite
to the force with which the Earth is attracting the pen,
even though the pen is much lighter than the Earth!
Newton also worked out that if you keep the masses of
the two bodies constant, the force of gravitational
attraction depends on the distance between their
centres:
mutual force
of attraction
Note that the gravitational force is larger
the closer the objects are together.
For any two particular masses, the gravitational force
between them depends on their separation as:
magnitude
of the
gravitational
force
between 2
fixed
masses, Fgrav
as the separation between the
masses is increased, the
gravitational force of attractions
between them decreases quickly.
distance between the masses, R
Your pen dropping to the floor and a satellite in orbit
around the Earth have something in common - they are
both in freefall.
To see this, remember Newton’s thought experiment
from the Activity Solar System Orbits:
On all these trajectories,
the projectile is in free fall
under gravity.
(If it were not, it would
travel in a straight line that’s Newton’s
First Law of Motion.)
If the ball is not given enough “sideways” velocity, its
trajectory intercepts the Earth
- that is, it falls to Earth eventually.
trajectories
which
complete
orbits,
OnOn
allthe
these
trajectories,
themake
projectile
is in free
fall.the
projectile is travelling “sideways” fast enough ...
… that as it falls, the Earth curves away underneath
it, and the projectile completes entire orbits
without ever hitting the Earth.
(c) Apparent Weightlessness in Orbit
This astronaut on a
space walk is also
in free fall.
The astronaut’s
“sideways” velocity
is sufficient to keep
him or her in orbit
around the Earth.
Let’s take a little time to answer the following question:
Why do astronauts in the
Space Shuttle in Earth
orbit feel weightless?
Some common misconceptions which become apparent
in answers to this question are:
(a) there is no gravity in space,
(b) there is no gravity outside the Earth’s atmosphere, or
(c) at the Shuttle’s altitude, the force of gravity is very small.
Click on each alternative to see why we claim
that it’s a misconception!
Then see if you agree with our explanation ...
In a spacecraft (like the Shuttle) in Earth orbit, astronauts
are in free fall, at the same rate as their spaceships.
On allisthese
That
why they
trajectories,
experience
the weightlessness:
projectile is in free
just
fall.
as
a platform diver feels while diving down towards a pool,
or a sky diver feels while in free fall.
In the next Module we’ll look at one Solar System
orbit in particular
- that of the Moon around the Earth.
On all these trajectories, the projectile is in free fall.
Image Credits
NASA: View of Australia
http://nssdc.gsfc.nasa.gov/image/planetary/earth/gal_australia.jpg
NASA: Halley in 1910
http://pds.jpl.nasa.gov/planets/gif/smb/hal1910.gif
NASA: Space Shuttle
http://lisar.larc.nasa.gov/LISAR/IMAGES/SMALL/EL-1994-00718.jpeg
Now return to the Module home page, and read
more about gravity in the Textbook Readings.
Hit the Esc key (escape)
to return to the Module 4 Home Page
(a) There is no gravity in space?
At face value, this statement doesn’t bear too much
examination, because Newton’s Law of Gravitation has
been applied right from its inception to the motion of the
Moon and planets - and they are in space.
When people make this assumption, perhaps what they
are really saying is that the sort of gravity which makes
us feel heavy only exists on planetary surfaces
- but Newton developed the Law in the first place by
realizing that gravity as it acts on Earth (e.g. on an apple)
is the same force as that which acts on the Moon
and planets.
Back to the alternative answers
(b) There is no gravity outside the Earth’s
atmosphere?
Like (a), at first glance this misconceptions seems naïve,
because Newton’s Law of Gravitation has been applied
right from its inception to the motion of the Moon and
planets - and they are in space.
However what this statement might really be revealing
is a link many people perceive between gravity and air:
in other words, the mistaken idea that gravity does not exist
in a vacuum - that air, in some way, makes things heavy.
In fact, air actually makes objects feel very, very slightly
lighter - like buoyancy in a tank full of water, except that
air is so much less dense than water that the effect is
not noticeable.
The misconception that links gravity and air shows up in
some science fiction movies too
- watch for the one where a Concorde-type plane
“mistakenly” ends up in Earth orbit. The passengers
inside the plane can walk around, with a bit of care, but
astronauts sent up to help them float weightlessly about
outside!
Back to the alternative answers
(c) At the Shuttle’s altitude, the force of
gravity is very small?
This statement sounds reasonable - after all, the Shuttle
is way out in space - until you check it with calculations.
In fact, compared to the radius of the Earth (6378 km), a
typical Shuttle altitude above the Earth’s surface of 200 km
or so is pretty negligible.
At that altitude, the force of gravity is only 5% less than
on the Earth’s surface.
Back to the alternative answers