Circular and Elliptical Orbit Geometries.
Satellite orbits follow either circular or elliptical paths based on Keplers' laws of motion. Intrinsic assumptions are
that the mass of the satellite is insignificant compared to mass of planet Earth and that there are no other
gravitational or secondary disturbances, the only attractor being a perfectly spherical planet Earth.
Low Earth Orbits (LEO) range from 300Km altitude to just a couple of thousand kilometers and are
predominantly circular. Geosynchronous Earth Orbits (GSO) are located at an altitude of 35786 kilometers
altitude with the special case of Geostationary Earth Orbit (GEO) located at the same altitude but directly above
the equator. Both GSO and GEO tend to be circular. Medium Earth Orbits (MEO) range between these two limits
and can be circular or elliptical. Transfer orbits between the various levels, eg, GTO (transfer from low earth orbit
to GEO) are highly elliptical. Satellite orbits can have a range of inclinations from equatorial to polar.
In all cases to analyse the geometry we take a simple approach of setting the axis to be in the plane of the orbit
with Earth at a focal point of the orbit.
The point on the ellipse closest to earth is defined
as the perigee with a radial distance from earth
focus set as RP.
The furthest point is defined as apogee with a
radial distance RA.
The semi-major axis (a) is an important measure
of the size of the ellipse, along with the distance
between center and focus (c) which determines
the eccentricity (e = c/a) .
The orbit obtained by an particular satellite is
determined by its velocity. A balance between
the angular momentum and acceleration of the
satellite and the gravitational attraction of Earth
will cause it to maintain a stable path.
Circular Orbit
For a circular orbit the satellite will maintain a constant velocity whose angular acceleration will exactly match the
gravitational acceleration for its altitude. For a given altitude (h) the satellite's radial position to the center of the
Earth will be R = h+RE , where RE is the radius of the Earth. At this point the gravitational acceleration due to
the earth will be
g=
GM
2
R
where G is the universal gravitational constant and M is the mass of Earth. Typically the value of GM is taken to
be 3.986005x1014 m3/s2
The angular acceleration produced by the Earth's attraction will be
a=ω2 R
where
ω is the angular velocity of the satellite which is related to the satellite velocity (V) as ω=V /R .
For a stable circular path the accelerations must be equal so that
GM
V2
2
=g=a=ω
R=
2
R
R
Thus the required satellite velocity for a stable circular orbit at a given radius will be
√
V= (
Elliptical Orbit
GM
)
R
If the velocity of the satellite at a given altitude doesn't match the above requirement for a circular orbit then the
resulting orbit will be elliptical. The velocity will no longer be constant but will change with radius from earth.
The motion is still controlled by conservation of angular momentum and acceleration laws. (Kepler's Laws).
Conservation of angular momentum implies that m ω where m is the mass of the satellite, is constant at all
points on the orbit. Taking perigee and apogee points gives the following balance,
m ω=m V P R P=m V A R A
VP RA
=
V A RP
Conservation of energy implies that the sum of kinetic energy and gravitational potential for the satellite is also
conserved along the path. Kinetic energy of the satellite is
gravitational field is
−m g R=−m
1
2
mV , its potential energy in the Earth's
2
GM
. Equating the energy at perigee and apogee points gives,
R
1
GM 1
GM
mV 2P−m
= mV 2A −m
,
2
RP 2
RA
1
1
V 2P −V 2A =2GM ( − ) .
RP RA
Applying the conservation of angular momentum result gives,
2
V 2P −V 2P
( )
RP
R
2
A
=2GM
√
V P = 2GM
(
1
1
−
RP RA
)
2
or
V 2A
( )
RA
R
2
P
−V 2A =2GM
( R1 − R1 )
P
√
A
RP
RA
1
1
or V A = 2GM
R P R A+ R P
R A R A+ R P
( )
( )
or rearranging for solution of radius gives,
RA=
RP
(
2GM
−1
RP V 2P
)
and
R P=
RA
(
2GM
−1
R A V 2A
)
The eccentricity (e) of the orbit can be calculated as
e=
RP V 2P
−1
GM
Given the conditions at a perigee or apogee, the shape of
the orbit can thus be determined accurately.
Period of Orbit.
For a circular orbit the period can be calculated from the
constant angular velocity.
2 π 2 π R 2 π R3 /2
P= ω =
=
V
GM
√
For an elliptical orbit the conservation laws allow a
mean motion approximation to be applied. Even though
the satellite velocity is increasing and decreasing along
the orbit, by simulating a circular orbit based on the
center of the ellipse, an equivalent period is obtained with the simulated satellite moving at a constant velocity
along this mean motion path. The period of the elliptical orbit will be
2 π a3 /2
P=
√GM
Location within an Orbit
Location of the satellite at any point in time within the orbit can be found by use of the mean motion simulation
and translation to a true anomaly ( ν ) (true angular location).
The process for calculating true anomaly (angular position) will be done in three steps. A mean anomaly (M) can
be predicted assuming the mean motion path for the satellite. Using the equal angle sweep equations based on
Kepler's Laws a relationship between M and eccentric anomaly (E) can be obtained, then using triangular
geometry as shown in the above figure true anomaly ( ν ) can be obtained from E.
The relationship between E and M is
M =E−e . sin(E)
Unfortunately this is a transcendental function and is not directly invertable, so approximate solutions or iterative
solutions are required to find E.
The exact relationship is shown in the following figure.
The geometric relationship between true anomaly and eccentric anomaly
√
1+ e
E
tan ν =
tan
2
1−e
2
( )
()
For small eccentricity, the relationship between true anomaly and mean anomaly can be approximated by,
ν≈ M + 2 e sin ( M )+ 1.25 e 2 sin(2M)
Alternate solutions to find satellite positions will require a numerical approach.
Numerical Integration for Orbit Simulation.
An alternate to the analytical solutions for orbit
prediction is a Euler time-stepping integration
technique using simple application of Newtons laws.
At any instant of time a satellite will experience an
acceleration that is due to the sum of all gravitational
effects applied to it. If it is assumed that the
acceleration is constant for a small time step then
changes to the satellites velocity and position can be
predicted. By continually updating the position and
recalculating the acceleration over many of these
small time steps a satellites trajectory can be mapped.
The following solution is based on the application of
this method to a simple orbital plane (two
dimensional).
At an initial point (x0,y0) the satellite can be assumed
to have x-dirn and y-dirn velocity components (u0,v0)
and will be subject to accelerations,
a x =−g .cos (θ)=−g . x0 / R
a y =−g . sin( θ)=−g . y 0 / R
where the local gravitational value will be
g=
GM
.
R2
Over a small time step ( Δ t ) the change in velocity can be approximated as :
u=u 0+ Δ t . a x and v =v 0 + Δ t . a y
The change in position will be :
x=x 0+ u 0. Δ t+
1
a x Δ t 2 and
2
y= y 0 + v 0. Δ t+
1
a y Δ t2
2
The new position can be used as the initial position for the next time step. The acceleration can be recalculated
for the new position and another step taken along the track. The process can continue in this mode for as many
time steps as is necessary to define a complete track.
A same MATLAB code is available to demonstrate this method : orbit.m
The advantage of a numerical method is that it is not limited to simple orbits and can include gravitational effects
from other objects or perturbations from other mechanisms that may cause acceleration or deceleration of the
satellite.
The disadvantage of the numerical method is its dependency on the length of the time step to ensure accuracy. In
many cases the time-step will need to be very small to ensure the assumption of constant acceleration is
maintained and a desired accuracy of solution achieved. The solution process can thus require a large number of
iterations and hence take a long time. Modifications to the constant acceleration Euler integration method are of
course quite plausible and may be used to improve solution efficiency.
m×
dV
= T −D−mg cos (θ)
dt
Rocket Launch/Boost/Trajectory Change Calculations
Initial launch parameters for a rocket can be estimates by analysing the following dynamic force
balance.
V = Velocity
T= Thrust
L= Lift
D=Drag
mg =weight
θ = angle between flight path and horizontal
Along the direction of the flight path
m×
dV
= T −D−mg cos (θ)
dt
Normal to the flight path we can assume Lift balances
residual weight.
L ≈ mg sin(θ)
Acceleration along the flight path will be due to thrust
of the rocket motors but also reduced by the
aerodynamic drag and the deceleration due to gravity.
Over a small period of time the change in vehicle velocity will be
dV =
T
D
. dt− . dt− g cos(θ). dt
m
m
To obtain overall changes in vehicle velocity the above expression will need to be integrated over the time of burn
( t b ) that the thrust is applied.
V
∫V
final
initial
dV = V initial −V final = Δ V =
tb
∫0
t D
t
T
. dt−∫0 . dt−∫0 g cos (θ). dt
m
m
b
b
In orbit there is no atmospheric drag and solar wind drag is minimal compared to thrust. For circular orbits θ is 90o
so cos (θ)=0 . For elliptical orbits the acceleration due to gravity speeds up and then slows down the rocket
in a balanced manner so for small period rockets burns, the elliptical orbit natural velocity change effect can be
taken out of the equation.
So for these simple cases
ΔV =
tb
T
∫o m . dt
=
T
dt
∫ m . dm . dm
=
m final
∫m
initial
T 1
. . dm
dm m
dt
dm
is the rate of change of mass of the vehicle and typically this is a reducing mass equal to the amount of
dt
exhaust gas leaving the exit of the nozzle.
−dm
= ṁ f Where ṁ f is the flow rate of exhaust product from the rocket.
dt
−T 1
. . dm
ṁ f m
Ideally thrust is produced by the exhaust momentum T = ṁ f . V e and by definition rocket motor efficiency
ΔV =
m final
∫m
initial
is measured by the amount thrust produced for a given mass flow rate of fuel , Specific Impulse (Isp)
Isp =
T
ṁ f g 0
or
T
= Isp.g 0 = V e
ṁ f
Hence
ΔV =
m final
∫m
initial
m
1
. dm = Isp.g 0. ∫m
m
−Isp. g 0 .
final
initial
−1
. dm assuming Isp is constant.
m
Δ V = − Isp. g 0. (ln(m final)−ln(minitial )) = Isp.g0. (ln( minitial )−ln (m final )) = Isp. g 0 . ln(
Δ V = Isp. g 0 . ln(
minitial
)
m final
minitial
m
) = V e ln( initial )
m final
m final
This result can be applied in orbit to determine the amount of fuel required for orbital manoeuvres. Ideally
neglecting gravity and drag it can be used to predict the amount of fuel used for launch..
However, how can the effects of drag and gravity be included in a launch calculation?
By definition
1
D=C D . ρV 2 A
2
–
C D – drag coefficient (may not always be constant)
ρ – density of fluid . Varies logarithmically with altitude.
V – velocity of vehicle
A – frontal cross-sectional area (may change as stages are released).
Gravitational attraction (g) changes with altitude and the flight path angle (θ) with change from zero to 90o as the
rocket goes from vertical at launch to a parallel path in low earth orbit.
Simple formulae and integrations for these non-linear parameters is not possible
tb
∫0
D
. dt
m
and
tb
∫0 g cos(θ). dt
The simplest approach again may be numerical rather than trying to find an analytical solution to be above
equations . Extending the previous time stepping approach and including the additional resistance forces would
allow prediction of accelerations at an individual point in time and hence a prediction of the velocity change and
new position a small time later.
a x =−g . cos (θ)−
Dx
,
m
a y =−g . sin(θ)−
Dy
m