Unit 3- Concept of orbit
3.1 Introduction
3.2 Two body problem and Kepler’s laws
3.3 Orbital parameters and orbit from initial condition
3.4 Types of orbits.
3.1 Introduction :The concept of an orbit refers to the path
that a celestial body, such as a planet or
moon, follows as it moves around a
larger body, like a star.
It is an essential concept for
understanding the behavior of celestial
bodies in the solar system and beyond.
Orbits can take various shapes, including
elliptical, circular, or irregular.
Elliptical orbits have an oval shape,
while circular orbits are perfectly round.
Irregular orbits do not follow predictable
patterns and can be influenced by the
gravitational forces of nearby celestial
bodies.
The motion of celestial bodies in an orbit is determined by the
balance between the body's forward momentum and the
gravitational pull of the larger body around which it is
orbiting.
This balance allows the celestial body to continually fall
towards the larger body while also moving forward fast
enough to avoid a collision.
This results in a stable, repetitive path known as an orbit.
Studying orbits is crucial for understanding celestial
mechanics, predicting astronomical phenomena, and enabling
space exploration.
By investigating the characteristics of orbiting bodies,
scientists can determine their properties such as distance,
speed, and gravitational interactions.
This concept plays a crucial role in understanding the motion and behavior of
celestial bodies within the solar system and beyond.
Orbits can take various shapes, including elliptical, circular, or irregular,
depending on the gravitational forces exerted by nearby celestial bodies.
The understanding of orbit is essential to comprehend how celestial bodies interact
and move in space.
 By studying orbits, scientists can predict and explain phenomena such as
planetary motion, lunar cycles, and the paths of comets and asteroids.
Orbits are governed by the principles described in Johannes Kepler's laws of
planetary motion, which state that planets follow elliptical paths with the central
body being located at one of the focal points.
Furthermore, the concept of orbit is instrumental in enabling space
missions and satellite deployments.
By carefully calculating orbital paths, scientists and engineers can
launch satellites into specific orbits to achieve desired functions, such
as communication, weather observation, or scientific research.
 Precise knowledge of orbits is crucial for spacecraft navigation and
ensures their successful operation in space.
In summary, the notion of orbit encompasses the path taken by a
celestial body as it revolves around a larger body.
It is a fundamental concept in understanding the dynamics of celestial
objects in our solar system and beyond, as well as guiding space
exploration and satellite technology.
3.2 Two body problem and Kepler’s laws :
The two-body problem in celestial mechanics is the problem of determining the
motion of two celestial bodies, typically stars or planets, under the influence of
their mutual gravitational attraction.
 It involves predicting the positions and velocities of the bodies at different points
in time, given their initial conditions.
Kepler's laws, formulated by Johannes Kepler in the 17th century, provide a
mathematical model for predicting the orbits of celestial bodies in the two-body
problem.
These laws are based on careful observations made by Kepler of the motion of the
planets and describe the shape, size, and relative distances between the bodies.
Kepler's First Law, also known as the law of ellipses, states that the orbits of the bodies
are elliptical, with one of the bodies located at one of the ellipse's foci.
This law describes the shape of the orbits and predicts that celestial bodies move in
elliptical paths rather than perfect circles.
Kepler's Second Law, also known as the law of equal areas, states that a line connecting
the two celestial bodies sweeps out equal areas in equal times.
This law describes the speed at which the bodies move in their orbits, with the bodies
moving faster when they are closer together and slower when they are farther apart.
Kepler's Third Law, also known as the harmonic law, relates the orbital periods of the
celestial bodies to their average distances from each other.
It states that the square of the orbital period is proportional to the cube of their average
distance.
This law allows astronomers to calculate the relative distances between celestial bodies
based on their observed orbital periods.
By using Kepler's laws, astronomers can calculate the positions of celestial bodies at
different points in time and study the dynamics of their orbits.
 This helps in understanding the motion of planets, satellites, and other celestial objects,
and has been instrumental in the development of celestial mechanics and our
understanding of the universe.
3.2 Two body problem and Kepler’s laws
What is the two-body problem in space?
The two-body problem is an astrodynamics model that considers
only two masses. One is usually a celestial body, and the other is
usually a spacecraft, whose motion
is of interest.
3.3 Orbital parameters and orbit from initial
condition
Orbital parameters are the set of characteristics
that define the motion of an object in space,
particularly in relation to its orbit around another
object. These parameters include the following:
Semi-major axis (a): This is the average distance
between the centers of the two objects. It is the
primary measure of the size of the orbit.
Eccentricity (e): This parameter describes the
shape of the orbit. It measures how elongated or
circular the orbit is.
e = 0 ; perfectly circular orbit,
e = 1 ; highly elongated or elliptical orbit.
Inclination (i): This is the angle between the orbital plane and the reference plane, which is often the
plane of the Earth's equator. It defines the tilt of the orbit with respect to the reference plane.
 The inclination indicates the tilt of an orbit. It is the angle measured between the K axis and the angular
momentum vector, h , as shown in the figure below.
Longitude of the ascending node
(Ω): This parameter defines the
position where the orbit passes
from below to above the reference
plane. It is the angle between the
reference direction (such as a
fixed star) and the ascending
node, which is the point where the
orbit crosses the reference plane
from below.
Argument of periapsis (ω): This parameter
determines the position of the object at its
closest approach to the primary object
(periapsis). It is the angle between the
ascending node and the periapsis point.
Mean anomaly (M): This parameter
indicates the object's position along its
orbit at a specific point in time.
These orbital parameters collectively define
the path and behavior of objects in space,
such as planets, moons, satellites, and other
celestial bodies. They are crucial in
understanding and predicting the motions of
objects in the solar system and beyond.
What is the formula for orbital parameters?
The orbit formula,
r=
𝑝
1+𝑒 cos 𝞱
=(
ℎ2
𝞵
) / ( 1 + e cos θ ) ,
Where, r = radius,
θ = angle between semi-major axis and position vector r.
p= (
𝐵
𝞵
ℎ2
𝞵
) = parameter,
e = = eccentricity.
This equation gives the position of body m2 in its orbit around m1 as a function of the
true anomaly. For many practical reasons, we need to be able to determine the position.
The value for e, the eccentricity, determines the shape the orbit will take. This table shows the
various values for e, and the resulting shape of the conic section:
e
Shape
e =0
Circle
e <1
Elipse
e =1
Parabola
e >1
Hyperbola
We can calculate e directly without having to calculate B first.
We can relate e to the energy and the angular momentum of the satellite by,
e=
2𝞮ℎ 2
𝞵2
+1
The period of an orbit can be given as,
T = 2𝞹
𝑎3
𝞵
Apses:
The line between the foci is known as the
major axis (in an ellipse and a hyperbola).
The points at the intersection of the curve and
the major axis are known as the apses.
The point nearest the primary focus is called
the periapsis, and the point nearest to the
secondary focus is called the apoapsis.
There are several other words that can be used
interchangeably
Central Body
Closest Distance
Furthest Distance
General
Periapsis/Pericenter
Apoapsis/Apocenter
Earth
Perigee
Apogee
Sun
Perihelion
Aphelion
Orbit initial conditions refer to the starting conditions of an object's orbit around a celestial
body, such as a planet or moon. These conditions include the object's initial position, velocity,
and angle of incidence relative to the celestial body. Understanding these initial conditions is
crucial for predicting the future trajectory of an object's orbit.
e.g. 1) The altitude of a satellite at perigee is 500 km and its orbital eccentricity is 0.1. Find:
a) The satellite’s altitude at apogee
b) The orbit’s specific mechanical energy, 𝞮
c) The magnitude of the orbit’s specific angular momentum, h
d) The satellite’s speed at apogee
Apo centre, Ra = a(1+e) = 7642(1+0.1)
a) The satellite’s altitude at apogee
Ra = 8406 km
Since
So, satellite’s altitude at
pericentre, Rp = a(1-e)
apogee = 8406 – 6378 km
6378 km + 500 km = a(1-0.1)
Altitude at apogee = 2028 km
a = 6878/0.9 km
a = 7642 km
and
b) The orbits specific mechanical energy 𝞮 is,
−𝞵
2𝑎
−398600.5 𝑘𝑚3/𝑠2
2(42241 𝑘𝑚)
since, 𝞮 =
𝞮=
= - 4.718
𝑘𝑚2
𝑠2
= - 26.08
𝑘𝑚2
𝑠2
c) Magnitude of orbits specific angular momentum,
h = 𝞵𝑎(1 − 𝑒2) = 398600.5 ∗ 7642 ∗ (1 − 0.1 2) = 54915 km2/s
d) Satellite speed at apogee,
h = 𝑅𝑎 𝑣𝑎
ℎ
𝑣𝑎 = 𝑣 = 6.53 km/s
𝑎
3.4. Types of Orbits :
 Orbits play a crucial role in space as they determine the path and trajectory of celestial objects and
satellites around a planet or celestial body.
 Understanding the different types of orbits is essential for various applications in Earth
observation, communication, navigation, and space exploration.
1) Geostationary orbit (GEO)
2) Low Earth orbit (LEO)
3) Medium Earth orbit (MEO)
4) Polar orbit and Sun-synchronous orbit (SSO)
5) Transfer orbits and geostationary transfer orbit (GTO)
6) Lagrange points (L-points)
1) Geostationary orbit (GEO) :Satellites in geostationary orbit (GEO)
circle Earth above the equator from west
to east following Earth’s rotation – taking
23 hours 56 minutes and 4 seconds – by
travelling at exactly the same rate as Earth.
 This makes satellites in GEO appear to be
‘stationary’ over a fixed position.
In order to perfectly match Earth’s
rotation, the speed of GEO satellites
should be about 3 km per second at an
altitude of 35 786 km.
This is much farther from Earth’s surface
compared to many satellites.
GEO is used by satellites that need to
stay constantly above one particular place
over Earth, such as telecommunication
satellites.
This way, an antenna on Earth can be
fixed to always stay pointed towards that
satellite without moving.
It can also be used by weather monitoring
satellites, because they can continually
observe specific areas to see how weather
trends emerge there.
Satellites in GEO cover a large range
of Earth so as few as three equallyspaced satellites can provide near
global coverage.
This is because when a satellite is this
far from Earth, it can cover large
sections at once.
This is akin to being able to see more
of a map from a meter away compared
with if you were a centimeter from it.
So to see all of Earth at once from
GEO far fewer satellites are needed
than at a lower altitude.
Low Earth orbit (LEO) :A low Earth orbit (LEO) is, as the name
suggests, an orbit that is relatively close to
Earth’s surface.
It is normally at an altitude of less than
1000 km but could be as low as 160 km
above Earth – which is low compared to
other orbits, but still very far above
Earth’s surface.
By comparison, most commercial
airplanes do not fly at altitudes much
greater than approximately 14 km, so even
the lowest LEO is more than ten times
higher than that.
Unlike satellites in GEO that must
always orbit along Earth’s equator, LEO
satellites do not always have to follow a
particular path around Earth in the same
way – their plane can be tilted.
This means there are more available
routes for satellites in LEO, which is one
of the reasons why LEO is a very
commonly used orbit.
LEO’s close proximity to Earth makes it
useful for several reasons.
It is the orbit most commonly used for
satellite imaging, as being near the
surface allows it to take images of higher
resolution.
It is also orbit used for International
Space Station (ISS), as it is easier for
astronauts to travel to and from it at a
shorter distance.
Satellites in this orbit travel at a speed
of around 7.8 km per second; at this
speed, a satellite takes approximately
90 minutes to circle Earth, meaning
the ISS travels around Earth about 16
times a day.
However, individual LEO satellites are
less useful for tasks such as
telecommunication, because they
move so fast across the sky and
therefore require a lot of effort to track
from ground stations.
Instead, communications satellites in
LEO often work as part of a large
combination or constellation, of multiple
satellites to give constant coverage.
In order to increase coverage,
sometimes constellations like this,
consisting of several of the same or
similar satellites, are launched together
to create a ‘net’ around Earth.
This lets them cover large areas of Earth
simultaneously by working together.
Ariane 5 carried its heaviest 20-tonne
payload, the Automated Transfer Vehicle
(ATV), to the International Space
Station located in low Earth orbit.
Medium Earth orbit (MEO) :Medium Earth orbit comprises a wide range of
orbits anywhere between LEO and GEO.
It is similar to LEO in that it also does not need to
take specific paths around Earth, and it is used by
a variety of satellites with many different
applications.
It is very commonly used by navigation satellites,
like the European Galileo system (pictured).
Galileo powers navigation communications
across Europe, and is used for many types of
navigation, from tracking large jumbo jets to
getting directions to your smartphone.
 Galileo uses a constellation of multiple satellites
to provide coverage across large parts of the
world all at once.
Polar orbit and Sun-synchronous orbit
(SSO) :Satellites in polar orbits usually travel
past Earth from north to south rather
than from west to east, passing
roughly over Earth's poles.
Satellites in a polar orbit do not have
to pass the North and South Pole
precisely; even a deviation within 20
to 30 degrees is still classed as a polar
orbit.
 Polar orbits are a type of low Earth
orbit, as they are at low altitudes
between 200 to 1000 km.
Sun-synchronous orbit (SSO) :is a particular kind of polar orbit. Satellites
in SSO, travelling over the polar regions,
are synchronous with the Sun.
This means they are synchronized to
always be in the same ‘fixed’ position
relative to the Sun.
 This means that the satellite always visits
the same spot at the same local time – for
example, passing the city of Paris every
day at noon exactly.
This means that the satellite will always
observe a point on the Earth as if
constantly at the same time of the day,
which serves a number of applications; for
example, it means that scientists and those
who use the satellite images can compare
how somewhere changes over time.
This is because, if you want to monitor an
area by taking a series of images of a
certain place across many days, weeks,
months, or even years, then it would not
be very helpful to compare somewhere at
midnight and then at midday – you need
to take each picture as similarly as the
previous picture as possible.
 Therefore, scientists use image series like
these to investigate how weather patterns
emerge, to help predict weather or storms;
when monitoring emergencies like forest
fires or flooding; or to accumulate data on
long-term problems like deforestation or
rising sea levels.
Often, satellites in SSO are synchronized
so that they are in constant dawn or dusk
– this is because by constantly riding a
sunset or sunrise, they will never have the
Sun at an angle where the Earth shadows
them.
A satellite in a Sun-synchronous orbit
would usually be at an altitude of between
600 to 800 km.
At 800 km, it will be travelling at a speed
of approximately 7.5 km per second.
Transfer orbits and geostationary transfer
orbit (GTO) :Transfer orbits are a special kind of orbit
used to get from one orbit to another.
 When satellites are launched from Earth
and carried to space with launch vehicles
such as Ariane 5, the satellites are not
always placed directly on their final orbit.
Often, the satellites are instead placed on
a transfer orbit: an orbit where, by using
relatively little energy from built-in
motors, the satellite or spacecraft can
move from one orbit to another.
This allows a satellite to reach, for
example, a high-altitude orbit like GEO
without actually needing the launch
vehicle to go all the way to this altitude,
which would require more effort – this is
like taking a shortcut.
 Reaching GEO in this way is an
example of one of the most common
transfer orbits, called the geostationary
transfer orbit (GTO).
Orbits have different eccentricities – a
measure of how circular (round) or
elliptical (squashed) an orbit is.
In a perfectly round orbit, the satellite is
always at the same distance from the
Earth’s surface – but on a highly
eccentric orbit, the path looks like an
ellipse.
On a highly eccentric orbit like this, the
satellite can quickly go from being very far
to very near Earth’s surface depending on
where the satellite is on the orbit.
In transfer orbits, the payload uses engines to
go from an orbit of one eccentricity to
another, which puts it on track to higher or
lower orbits.
After liftoff, a launch vehicle makes its way
to space following a path shown by the
yellow line, in the figure.
At the target destination, the rocket releases
the payload which sets it off on an elliptical
orbit, following the blue line which sends the
payload farther away from Earth.
The point farthest away from the Earth on the
blue elliptical orbit is called the apogee and
the point closest is called the perigee.
When the payload reaches the apogee at
the GEO altitude of 35 786 km, it fires its
engines in such a way that it enters onto
the circular GEO orbit and stays there,
shown by the red line in the diagram.
So, specifically, the GTO is the blue path
from the yellow orbit to the red orbit.
Lagrange points :For many spacecraft being put in
orbit, being too close to Earth can
be disruptive to their mission – even
at more distant orbits such as GEO.
For example, for space-based
observatories and telescopes whose
mission is to photograph deep, dark
space, being next to Earth is hugely
detrimental because Earth naturally
emits visible light and infrared
radiation that will prevent the
telescope from detecting any faint
lights like distant galaxies.
Photographing dark space with a
telescope next to our glowing
Earth would be as hopeless as
trying to take pictures of stars
from Earth in broad daylight.
 Lagrange points, or L-points,
allow for orbits that are much,
much farther away (over a million
kilometers) and do not orbit Earth
directly.
These are specific points far out in space
where the gravitational fields of Earth and
the Sun combine in such a way that
spacecraft that orbit them remain stable
and can be ‘anchored’ relative to Earth.
If a spacecraft was launched to other
points in space very distant from Earth,
they would naturally fall into an orbit
around the Sun, and those spacecraft
would soon end up far from Earth, making
communication difficult.
Instead, spacecraft launched to these
special L-points stay fixed, and remain
close to Earth with minimal effort without
going into a different orbit.
The most used L-points are L1 and L2.
These are both four times farther away from
Earth than the Moon – 1.5 million km,
compared to GEO’s 36 000 km – but that is
still only approximately 1% of the distance of
Earth from the Sun.
Many ESA observational and science
missions were, are, or will enter an orbit
about the L-points.
For example, the solar telescope SOHO and
LISA Pathfinder at the Sun-Earth L1 point;
Herschel, Planck, Gaia, Euclid, Plato, Ariel,
JWST, and the Athena telescope are or will
be at the Sun-Earth L2 point.
1) What are the orbital parameters used for
positioning a satellite?
The five parameters a ,e ,I ,Ω ,
ω completely define the satellite orbit in
space and the sixth parameter M will define
motion of satellite in orbit. a and e give
shape of ellipse. i and Ω relates orbital
plane position with respect to earth's
equatorial plane