# Circular and Elliptical Orbit Geometries.

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1 Circular and Elliptical Orbit Geometries. Satellite orbits ollow either circular or elliptical paths based on Keplers' laws o motion. Intrinsic assumptions are that the mass o the satellite is insigniicant compared to mass o planet Earth and that there are no other gravitational or secondary disturbances, the only attractor being a perectly spherical planet Earth. Low Earth Orbits (LEO range rom 300Km altitude to just a couple o thousand kilometers and are predominantly circular. Geosynchronous Earth Orbits (GSO are located at an altitude o kilometers altitude with the special case o 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. Transer orbits between the various levels, eg, GTO (transer rom low earth orbit to GEO are highly elliptical. Satellite orbits can have a range o inclinations rom equatorial to polar. In all cases to analyse the geometry we take a simple approach o setting the axis to be in the plane o the orbit with Earth at a ocal point o the orbit. The point on the ellipse closest to earth is deined as the perigee with a radial distance rom earth ocus set as. The urthest point is deined as apogee with a radial distance. The semi-major axis (a is an important measure o the size o the ellipse, along with the distance between center and ocus (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 o the satellite and the gravitational attraction o 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 or its altitude. For a given altitude (h the satellite's radial position to the center o the Earth will be R = h+r E, where R E is the radius o the Earth. At this point the gravitational acceleration due to the earth will be g= GM R 2 where G is the universal gravitational constant and M is the mass o Earth. Typically the value o GM is taken to be x0 4 m 3 /s 2 The angular acceleration produced by the Earth's attraction will be a=ω 2 R where ω is the angular velocity o 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 R 2 =g=a=ω2 R= V 2 R

2 Thus the required satellite velocity or a stable circular orbit at a given radius will be Elliptical Orbit V = (GM R I the velocity o the satellite at a given altitude doesn't match the above requirement or a circular orbit then the resulting orbit will be elliptical. The velocity will no longer be constant but will change with radius rom earth. The motion is still controlled by conservation o angular momentum and acceleration laws. (Kepler's Laws. Conservation o angular momentum implies that m ω where m is the mass o the satellite, is constant at all points on the orbit. Taking perigee and apogee points gives the ollowing balance, m ω=m V P =m V A V P = V A Conservation o energy implies that the sum o kinetic energy and gravitational potential or the satellite is also conserved along the path. Kinetic energy o the satellite is 2 mv 2, its potential energy in the Earth's gravitational ield is m gr= m GM R. Equating the energy at perigee and apogee points gives, 2 mv 2 P m GM = 2 mv 2 A m GM, V P 2 V A 2 =2GM (. Applying the conservation o angular momentum result gives, V 2 2 P V P ( R 2 P 2 =2GM ( or = V 2GM P ( V 2 A( + or V A = R 2 A 2 ( 2GM or rearranging or solution o radius gives, = ( 2GM V 2 P and The eccentricity (e o the orbit can be calculated as 2 e= R V P P GM V 2 A=2GM ( = ( 2GM V 2 A + Given the conditions at a perigee or apogee, the shape o the orbit can thus be determined accurately. Period o Orbit. For a circular orbit the period can be calculated rom the constant angular velocity. P= 2π ω = 2π R 2π R3/2 = 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

3 center o the ellipse, an equivalent period is obtained with the simulated satellite moving at a constant velocity along this mean motion path. The period o the elliptical orbit will be /2 2 π a3 P= GM Location within an Orbit Location o the satellite at any point in time within the orbit can be ound by use o the mean motion simulation and translation to a true anomaly ( ν (true angular location. The process or calculating true anomaly (angular position will be done in three steps. A mean anomaly (M can be predicted assuming the mean motion path or 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 igure true anomaly ( ν can be obtained rom E. The relationship between E and M is M=E e. sin(e Unortunately this is a transcendental unction and is not directly invertable, so approximate solutions or iterative solutions are required to ind E. The exact relationship is shown in the ollowing igure.

4 The geometric relationship between true anomaly and eccentric anomaly tan ( ν 2 = + e e tan ( E 2 For small eccentricity, the relationship between true anomaly and mean anomaly can be approximated by, ν M+ 2esin( M+.25 e 2 sin(2m Alternate solutions to ind satellite positions will require a numerical approach. Numerical Integration or Orbit Simulation. An alternate to the analytical solutions or orbit prediction is a Euler time-stepping integration technique using simple application o Newtons laws. At any instant o time a satellite will experience an acceleration that is due to the sum o all gravitational eects applied to it. I it is assumed that the acceleration is constant or 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 o these small time steps a satellites trajectory can be mapped. The ollowing solution is based on the application o this method to a simple orbital plane (two dimensional. At an initial point (x 0,y 0 the satellite can be assumed to have x-dirn and y-dirn velocity components (u 0,v 0 and will be subject to accelerations, a x = g.cos(θ= g. x 0 /R a y = g.sin(θ= g. y 0 / R

5 where the local gravitational value will be g= GM R 2. 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+ 2 a x Δ t 2 and y= y 0 + v 0. Δ t+ 2 a yδ t 2 The new position can be used as the initial position or the next time step. The acceleration can be recalculated or the new position and another step taken along the track. The process can continue in this mode or as many time steps as is necessary to deine a complete track. A same MATLAB code is available to demonstrate this method : orbit.m The advantage o a numerical method is that it is not limited to simple orbits and can include gravitational eects rom other objects or perturbations rom other mechanisms that may cause acceleration or deceleration o the satellite. The disadvantage o the numerical method is its dependency on the length o the time step to ensure accuracy. In many cases the time-step will need to be very small to ensure the assumption o constant acceleration is maintained and a desired accuracy o solution achieved. The solution process can thus require a large number o iterations and hence take a long time. Modiications to the constant acceleration Euler integration method are o course quite plausible and may be used to improve solution eiciency. m dv = T D mg cos(θ

6 Rocket Launch/Boost/Trajectory Change Calculations Initial launch parameters or a rocket can be estimates by analysing the ollowing dynamic orce balance. V = Velocity T= Thrust L= Lit D=Drag mg =weight θ = angle between light path and horizontal Along the direction o the light path m dv = T D mg cos(θ Normal to the light path we can assume Lit balances residual weight. L mg sin(θ Acceleration along the light path will be due to thrust o the rocket motors but also reduced by the aerodynamic drag and the deceleration due to gravity. Over a small period o time the change in vehicle velocity will be dv = T m. D. g cos(θ. m To obtain overall changes in vehicle velocity the above expression will need to be integrated over the time o burn ( t b that the thrust is applied. V inal t b T V dv = V initial V inal = Δ V = initial 0 m. t b D 0 m. t b 0 g cos(θ. In orbit there is no atmospheric drag and solar wind drag is minimal compared to thrust. For circular orbits θ is 90 o 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 or small period rockets burns, the elliptical orbit natural velocity change eect can be taken out o the equation. So or these simple cases Δ V = o t b T m. = T m. dm.dm = m inal minitial T. dm m.dm dm is the rate o change o mass o the vehicle and typically this is a reducing mass equal to the amount o exhaust gas leaving the exit o the nozzle. dm = m Where m is the low rate o exhaust product rom the rocket. m inal T Δ V = minitial. m m.dm Ideally thrust is produced by the exhaust momentum T = m. V e and by deinition rocket motor eiciency is measured by the amount thrust produced or a given mass low rate o uel, Speciic Impulse (Isp Isp = T T or = Isp.g m g 0 m 0 = V e

7 Hence m inal m inal Δ V = minitial Isp. g 0. m. dm = Isp.g 0. minitial m. dm assuming Isp is constant. Δ V = Isp. g 0. (ln(m inal ln(m initial = Isp.g 0. (ln(m initial ln (m inal = Isp. g 0.ln( m initial m inal Δ V = Isp. g 0.ln( m initial = V m e ln( m initial inal m inal This result can be applied in orbit to determine the amount o uel required or orbital manoeuvres. Ideally neglecting gravity and drag it can be used to predict the amount o uel used or launch.. However, how can the eects o drag and gravity be included in a launch calculation? By deinition D=C D. 2 ρv 2 A C D drag coeicient (may not always be constant ρ density o luid. Varies logarithmically with altitude. V velocity o vehicle A rontal cross-sectional area (may change as stages are released. Gravitational attraction (g changes with altitude and the light path angle (θ with change rom zero to 90 o as the rocket goes rom vertical at launch to a parallel path in low earth orbit. Simple ormulae and integrations or these non-linear parameters is not possible 0 t b D m. and t b 0 g cos(θ. The simplest approach again may be numerical rather than trying to ind an analytical solution to be above equations. Extending the previous time stepping approach and including the additional resistance orces would allow prediction o accelerations at an individual point in time and hence a prediction o the velocity change and new position a small time later. a x = g. cos(θ D x m, a = g. sin(θ D y y m

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