MATHEMATICAL MODEL OF THE

MOTION OF A SATELLITE LAUNCH

VEHICLE FROM LAUNCH TILL ORBIT

(AN ANALYSIS)

[9]

Varun Suriyanarayana Guided by Dr. Ranganath Navalgund

CONTENTS INTRODUCTION ....................................................................................................................................... 3

WHAT IS A SATELLITE? ............................................................................................................................ 3

WHAT IS A ROCKET? ................................................................................................................................ 4

FORCES AFFECTING THE LAUNCH VEHICLE OF THE SATELLITE(ROCKET) ................................................ 5

DESCRIPTION OF EACH FORCE ................................................................................................................ 5

MODELLING THE PROCESS BY WHICH A SATELLITE IS PLACED INTO ORBIT BY ITS LAUNCH

VEHICLE(ROCKET) BY WRITING EQUATIONS ........................................................................................... 6

DERIVATION OF EQUATIONS FOR ROCKET MOTION .............................................................................. 6

EQUATION FOR THRUST ......................................................................................................................... 6

TSIOKOVOLSKY ROCKET EQUATION........................................................................................................ 7

EQUATION FOR LINEAR MOTION ............................................................................................................ 8

CONCLUSION ......................................................................................................................................... 10

BIBLIOGRAPHY ...................................................................................................................................... 12

INTRODUCTION Satellites are a product of path breaking advances in technology which has resulted in huge benefits

to mankind. The first satellite ever launched was the Sputnik 1, which the USSR launched in 1957. [5]

Since then, over 6600 satellites have been launched and approximately 3600 are in orbit. [6] Of these

3600 satellites, approximately 1000 are operational. [5]

The satellite is taken into orbit by its corresponding launch vehicle, a rocket. In order to understand

and model its motion until it is placed into orbit, one must treat the whole system as a rocket which

operates on the principle of the law of conservation of momentum. [1] [2]

Satellites serve a variety of military and civilian purposes. On the military front, satellites serve the

purposes of defence, spying and when necessary launching and detecting missiles. The civilian

purposes include weather forecasting, mapping, communication and navigation. Satellites are also

used for the purpose of research. They orbit the earth trying to gather data pertinent to our own

planet. One such example is environmental data such as temperatures which can often be measured

by satellites. Satellites also detect changes in environmental composition of a large space over a long

period of time effectively, due to their location. In addition, some satellites carry parts of space

stations which are assembled in phases over a period of time. These space stations are used to

understand the effect of the earth’s gravitational field on various phenomena. [7]

The usefulness and importance of satellites leads to a fundamental question-How do they get into

orbit and how can we use physics to explain and model this process.

To understand this process one must first understand what a satellite is and what its launch vehicle,

a rocket is. Only then can one mathematically model the motion of the system

WHAT IS A SATELLITE? A satellite is defined as “a moon, planet or machine that orbits a planet or star.”[1] There are two

kinds of satellites-Natural satellites and Artificial satellites. The Natural satellites include celestial

bodies such as the planets revolving around the sun and moons revolving around planets. It can also

extend to include asteroids in orbit around a planet. The earth’s most well-known natural satellite is

the moon. Artificial satellites are all man-made devices that have been sent into orbit. This includes

all satellites that are currently in orbit and all the debris that has resulted from their operations

including collisions with other satellites. Therefore even dysfunctional satellites are still artificial

satellites.

Space probes which are not in orbit but are sent into space for the purpose of investigating other

planets are not considered as Artificial Satellites. Often, launching gear falls back into the

atmosphere and disintegrates or reaches the earth. In neither case do these count as satellites.

Space probes that orbit other celestial bodies like the sun or other planets are satellites of the sun or

those planets, not the earth. Similarly, objects that were in orbit around the earth but are no longer

in orbit, possibly because of larger gravitational forces due to other planets, are not satellites

although they were satellites when they were orbiting around the Earth. [1]

[3]

WHAT IS A ROCKET? A cylindrical projectile that can be propelled to a great height or distance by the combustion of its contents. [2] The acceleration required for launching a satellite is provided by a rocket engine. The rocket provides a force greater than the weight of the rocket. This allows it to overcome the force due to gravity and go upwards. The rocket operates on a principle known as the law of conservation of momentum. The rocket has a fuel which is called the propellant, a combustion chamber and the exhaust. The propellant can be liquid hydrogen, solid propellants, earth storable liquid propellants and other cryogenic fuels. There is also research being conducted in the area of semi-cryogenic fuels. Upon combustion, a large amount of energy is released. This heats up the gas giving it a greater kinetic energy and hence a greater velocity. This velocity is directed towards the earth. However, because the rocket has lost mass and the lost mass has a velocity towards the earth it must gain a velocity upwards and hence accelerate upwards.

[4]

FORCES AFFECTING THE LAUNCH VEHICLE OF THE

SATELLITE(ROCKET) The forces acting on the launch vehicle of the satellite are gravity, air resistance and the force from

the release of exhaust gasses (thrust). [8] [8] [10] [11] [12] [13] All of these forces combine to produce a

function of acceleration which in turn can be used to find functions for speed and distance from

which one can identify values of mass, energy, efficiency etc. However in order to do this one must

understand each of these forces and how each force works.

DESCRIPTION OF EACH FORCE Gravity- Since the distances over which a satellite must travel are significant and the strength of the

earth’s gravitational field decreases with distance, the simple F=mg is not quite appropriate. We

must use the equation from Newton’s law of gravitation, F= GMm/r2 where F is the force, G is the

universal gravitation constant, M is the mass of the earth, m is the mass of the rocket including the

propellant and the satellite r is the distance from the centre of the earth.

Air resistance-The force of air resistance acts against the direction of motion. It is defined by the

equation F=bvn. b is the coefficient of drag and depends on the shape of the rocket whilst n

determines how the force varies with velocity. The coefficient of drag is dependent on air density

which is not a constant. Therefore one must use a function of air density with height from the centre

of the earth. The domain should begin at height = the height of the surface of the earth, for which

sea level should be used.

Thrust-This is the force that propels the launch vehicle of the satellite into space. This force must

overcome the forces of gravity and air resistance and is provided by the release of exhaust gasses.

This force arises due to the law of conservation of momentum. As the launch vehicle of the satellite

loses mass and this mass accelerates towards the earth, an equal and opposite gain in the

momentum of the launch vehicle of the satellite is observed and since the mass is decreasing, by the

equation p=mv the velocity must increase. p is momentum of the launch vehicle of the satellite, m is

the mass of the launch vehicle of the satellite and v is the velocity of the rocket.

MODELLING THE PROCESS BY WHICH A SATELLITE IS

PLACED INTO ORBIT BY ITS LAUNCH VEHICLE(ROCKET)

BY WRITING EQUATIONS In order to model the motion of the launch vehicle carrying the satellite one has to consider the

various forces acting on it. From these forces one can understand the motion of the rocket at various

stages. From the equations of motion one can derive formulae for energy and mass. One factor that

has not been considered is relativistic effects because the maximum speed, although significantly

larger than speeds that we are used to on earth, is not comparable to the speed of light (satellites

and their launch vehicles can attain speeds of the order of 103 ms-1 to 104 ms-1 while the speed of

light is 3 X 108 ms-1 which is more than 10,000 times greater.

The equations for motion will be broken down into various parts, first the rocket equations will be

derived and then they will be applied to define the motion of the rocket at t=0 and for when the

rocket is on its way to the required orbit level. Note that the final stage, when the satellite is in orbit

does not require any of these equations in that there is no effect of air resistance or thrust. Only

gravity acts on it and thus its motion and energy can be expressed by applying Newton’s law of

gravitation.

DERIVATION OF EQUATIONS FOR ROCKET MOTION The equations for thrust, mass as a function of time, velocity as a function of time, distance as a

function of time, the burnout time and the burnout velocity provide analysts who are designing the

rocket with tools to determine which rocket, what propellant and how much payload must be

included in the launch vehicle and the satellite itself. They also provide scientists who are observing

the launch vehicle’s path an idea of how close to or far from the ideal position, velocity and mass the

rocket actually is and what correction measures must be taken for the particular space mission as

well as similar future missions.

EQUATION FOR THRUST The thrust required by a rocket to accelerate is greater than the weight of the rocket. However, in

order to escape the atmosphere as quickly as possible and minimise the effects of gravity and air

resistance as quickly as possible it is essential to accelerate the rocket as fast as possible. However, if

it accelerates too fast, the air resistance becomes too strong and causes the rocket to disintegrate.

Therefore an optimal thrust: weight ratio is 5:1.

In infinitesimal time:

Initial momentum of rocket = Mivi

Momentum of rocket after time dt: (Mi+dm)(vi+dv)

Momentum of exhaust gas after time dt: -dm(vg)

Miv= Miv + Mdv + vidm + dmdv - dm(vg)

Divding by dt: Mdv/dt + vdm/dt + dmdv/dt - vg(dm/dt)=0

Ma +(v +dv -vg)(dm/dt)= 0

Ma = -(v+ dv -vg)(dm/dt)

Thrust = -(v+ dv - vg)(dm/dt)

Since -(v+dv - vg) = velocity of the exhaust gasses relative to the rocket, if velocity of the exhaust

gasses relative to the rocket = vexhaust and dm/dt = R which is a constant, thrust can be re-written as:

Thrust= vexhaustR[9] [10] [12]

TSIOKOVOLSKY ROCKET EQUATION The equation for thrust can in turn can be used to find the Tsiokovolsky rocket equation which

connects the mass of the rocket to the change in its velocity in order to understand what the

efficiency of energy conversion is and how much energy is being produced.

M(dv/dt) = +(v + dv -vg)(dm/dt)

dv = (vexhaust)(dm/M)

Integrating both sides one gets:

∫vvf dv = (vexhaust)∫m0

mf 1/M dm

vf – v = (vexhaust)(ln(m0/mf))

vf = (vexhaust)(ln(m0/mf)) + v

Δv = (vexhaust)(ln(m0/mf))

This equation can be written in terms of a quantity referred to as specific impulse, commonly denoted as Isp. It is has two definitions, one which means that it measures impulse per unit mass, the more common one measures the impulse per unit weight where weight is g (standard gravity at the surface of the earth) times mass. In the first case, Isp =Thrust/ R whilst in the second case, it is defined to be Isp=Thrust/Rg.

In the second case, the Tsiokovolsky rocket equation can be written as:

Δv = Ispg(ln(m0/mf))[12]

EQUATION FOR LINEAR MOTION

The rocket’s burnout velocity and burnout time are the important aspects of modelling the launch vehicle’s motion. The burnout velocity and time express when the rocket is out of fuel and what its velocity is at that moment in time. [12] It is essential to optimise this value because if it is too much, the satellite can spiral away into outer space after the rocket has detached and if it is too little, the satellite will spiral into the earth after the rocket has been detached and the corresponding impact could be disastrous. This section will provide equations for these and thereby provide a model for the launch vehicle’s motion from the surface of the earth till the required orbit. Also given below is an equation for the distance a rocket has travelled as a function of time. This equation however is a second order differential equation and cannot be solved analytically. The equation has been presented in various forms which show that an analytical solution in this case is not possible.

This equation however neglects both air resistance and gravity. If one is to factor both in one reaches the equation:

mdv/dt = md2r/dt2 = Rvexhaust – b(dr/dt)2 – GMm/r2 = Rvexhaust - b(dr/dt)2 – GM(m0 – tdm/dt)/r2

d2r/dt2 = Rvexhaust/(m0 – tR) – b(dr/dt)2/(m0 – tR) – GM/r2

dr/dt = -vexhaustln(m0 – tR) - b∫(1/(m0 – tR))(dr)(dr/dt) – GM∫1/r2dt + c

r + b∫∫(1/(m0 – tR)) dr dr + GM∫∫1/r2dt dt = -vexhausttln(m0-tR) + (vexhaust/R)(t+m0ln(m0-tR)+d) + ct

In order to calculate burnout velocity and time

But now having an external force of gravity, mg

Solving for dv gives

Or as an equation of motion

Integrating over v, t and m (Where tbo is the burn-time of the rocket)

Gives an equation for the velocity at burnout

The term -gt is referred to as gravity loss. This represents the losses endured by launching in a gravity well. To maximize burnout velocity you want to minimize gravity

becomes useful to rewrite this equation in terms of a new parameter, thrust-to-weight ratio. We define thrust-to-weight ratio, Ψ, as the thrust (which we assume is constant) divided by the weight at liftoff, m0g. loss, which means burning the fuel as fast as possible. This makes sense because when you spend a long time burning fuel you are wasting energy lifting unburnt fuel to a higher altitude rather than your payload.

Practically, rocket motors are usually categorized by thrust and not burn time. So it

If we realize that thrust is

then

We can find a relation to the time it takes to burn through the fuel. If we take the burn rate to be constant (again, not a bad assumption) then the time is to burn out is

However we see that mf - m0 is negative! This okay because m-dot is also negative. What we actually want is the fuel mass divided by a (positive) burn rate.

We also want this in terms of mass ratio and thrust-to-weight ratio.

Now if we multiply by "one" we get

We recognize the term as the inverse of Ψ Thus we have the burnout time in reasonably terms

Plugging this in for t in our burnout velocity equation gets

Now we can also introduce the symbol μ for the mass ratio and replace ve with gIsp

[12]

CONCLUSION One must therefore conclude that while it is possible to mathematically model the motion of a

satellite and its launch vehicle from launch till orbit, it is beset by some inherent limitations. It is

possible to develop the model so as to estimate burnout time and velocity which are crucial factors

when making the launch vehicle for the satellite, deciding what kind of fuel and how much of it will

be needed . Although the model serves the purpose of explaining, through the use of equations,

how a rocket system takes a satellite up into orbit, the model is not comprehensive enough to

account for certain critical and related factors.

The first and most important factor is the presence of air resistance. Its importance is best explained

in projectile motion where air can reduce the distance a projectile travels by a factor of

approximately 10. Therefore for a model to be comprehensive, it must factor in air resistance. In

order to do this one must solve a second order differential equation. However, only first order

differential equations can be solved analytically. To use the second order differential equation, one

must use numeric methods which are too complicated and time consuming to do manually and

therefore supercomputers are used. This second order differential equation has been represented

above. Another essential point to note is that air resistance depends on the density of air which in

turn is related to air pressure. As altitude increases, the density of air reduces resulting in the

decrease of air resistance to the extent that satellites in higher orbits do not have any air around

them. Therefore an accurate model must factor this as well.

The second factor is that as the satellite and its launch vehicle ascend vertically, the value of g, the

gravitational field strength decreases. This decrease is related to 1/r2 where r is the distance from

the centre of the earth. Since the distances that a satellite and its launch vehicle must travel are of

the order of 7 times the earth’s radius, the value of g decreases by a factor of almost 50 during the

course of its journey. This means that the value of g decreases from 9.81 ms-2 to less than 0.2 ms-2. If

it is assumed to be 9.81 ms-2 throughout, the calculations based on the model above would suggest

that a satellite launch vehicle needs more and higher energy fuel than it actually needs and if this

extra fuel is used, the satellite and its launch vehicle could continue into outer space. Alternatively,

its higher than calculated velocity could cause a collision with another satellite and damage both

significantly while liberating space debris that could damage current and future satellites. There are

two other aspects of gravity that have been neglected. The first is the variation in g due to the

earth’s surface not being uniform and its density varying throughout however, given the distances

involved, the impact of this is negligible. The second is the gravitational force of other celestial

bodies. Although this may be of greater importance than the variation in density and lack of

uniformity of the earth’s surface, given that the satellites are far closer to the earth than any other

celestial body, this is also something that is often insignificant.

The third factor is the need for the rocket to change angle of flight with respect to the vertical

because the satellite must be placed into orbit with a velocity perpendicular to the vertical. This

necessity and its implication that some of the rocket’s thrust must be horizontal has also been

neglected in this model. Naturally, in order for a model to be useful and accurate this horizontal

thrust must be recognised and calculated.

The fourth factor is presence of lift force. This force acts perpendicular to the flow of air and is due

to a difference in air pressure at various points on the launch vehicle of the satellite. This is most

significant when the rocket is moving at an angle as opposed to straight upwards. As mentioned

earlier, practical rockets must slowly tilt because they need to place the satellite in orbit with the

requisite horizontal velocity.

The fifth factor that is overlooked is that rockets often do not follow the prescribed path and have

deviations. Therefore a complete model would analyse the probability and magnitude of such

deviations as well as provide a probabilistic estimate as to how much correction fuel the rocket

should have. The same also applies to the satellite itself once it is placed into orbit. If the satellite

deviates from its required path, it must have the fuel required to correct its path. A truly useful

model must provide values related to this and therefore account for this possibility.

BIBLIOGRAPHY [1] http://www.nasa.gov/audience/forstudents/5-8/features/what-is-a-satellite-58.html 15 July 2014

[2] http://encyclopedia2.thefreedictionary.com/rocket 9 August 2014

[3] http://www.spacetoday.org/images/Sats/MilSats/DSCS_SatInSpaceLockheedMartin.jpg 9 August

2014

[4] http://img.dictionary.com/rocket-145671-400-307.jpg 9 August 2014

[5] http://www.universetoday.com/42198/how-many-satellites-in-space/ 9 August 2014

[6] http://seattletimes.com/html/nationworld/2022236028_apxfallingsatellite.html 9 August 2014

[7] http://satellites.spacesim.org/english/function/index.html 9 August 2014

[8] http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/lecture-

notes/supplement8.pdf 18 July 2014

[9] http://www.real-world-physics-problems.com/rocket-physics.html 26 July 2014

[10] http://hyperphysics.phy-astr.gsu.edu/hbase/rocket.html 20 July 2014

[11] http://www2.estesrockets.com/pdf/Physics_Curriculum.pdf 31 July 2014

[12]http://www.braeunig.us/space/basics.htm 29 July 2014

[13]http://web.mit.edu/16.00/www/aec/rocket.html 29 July 2014