Planetary Motion Basics

Planetary System & Concept of Orbits

Kepler’s Laws of Planetary Motion

General Motion & Force Field in SpaceGeneral Motion & Force Field in Space

n- Body Problem Formulation

General Solution of Governing Equations

Simplified Two Body Problem

End point of an ascent mission is usually the start point

for the space motion mission of the spacecraft, as

brought out by the figure below.

Spacecraft Motion Basics

Payload Separation & Orbit Setup

Spacecraft Motion Basics

Upon injection spacecraft, which possesses both PE &

KE, starts moving on its own as per its initial conditions

and various forces and moments experienced by it.

There is, thus, a need to create mathematical models,

forces and motion variables tothat include the forces and motion variables, in order to

predict motion of spacecraft from this point onwards.

As spacecrafts have to operate in our solar system, it is

useful to understand the nature of motion of planets

within our own solar system, in order to create

applicable mathematical models.

Historical Background

Let us consider the developments which have strongly

influenced the motion models for orbital mechanics.

Ptolemaic Model (Earth

as Universe Centre).

Earliest approximation

of marginally ellipticof marginally elliptic

orbit of Earth.

Historical Background

Copernicus in 1543, recognized sun as the centre and

rearranged the orbit, which was still mainly a circle.

Galileo proved it through his telescope.

Kepler’s Formulation

Kepler, a contemporary of Galileo, formulated the laws

for planetary motion, based on observations of the solar

system, carried out by Tycho, the greatest astronomer.

Keppler’s Triangulation

Method. (Using Mars

orbit data)

Kepler’s Laws

Kepler formulated the laws for planetary motion, which

represent the fundamental mathematical relations in

Orbital Mechanics. The laws are as follows.

1. All planets move around the sun in planar and

elliptic orbits, with sun as one focus.elliptic orbits, with sun as one focus.

2. Radius vector from sun centre to planet centre

sweeps equal areas in equal time.

3. (Time period of the orbit)2 ∝ (Semi-major axis)3

These laws are generic and can be applied to any orbit,

including Earth-satellite system.

Orbital Solution Features

As can be seen, orbital solution, as proposed by Kepler,

is based on the premise that all objects in space will

form an orbit, which can be observed.

However, there are no explicit guarantees in Kepler’s

laws that all objects left in space, will form orbits.laws that all objects left in space, will form orbits.

This issue was settled by Newton, who formally derived

the Kepler’s laws, using his own three laws of motion.

This also overruled the Descartes’ vortex theory for

planetary motion. This has led to the evolution of

particle motion theory, which is central to the

formulation of orbital mechanics.

General Motion in Space

As other solar systems are at great distances from our

sun, most models of motion of planets consider only sun.

Force Field in Space

In outer space, there is practically no atmosphere and

therefore no forces related to air are present.

Also, spacecraft has limited propulsion capability, which

is to be used only for manoeuvres, resulting in no

propulsive forces.propulsive forces.

Forces due to magnetic field and solar radiation are

very small and have impact only on the long term drift

of the satellite from its path.

Thus, gravity is the only important force, which of

course is an internal force, as a result of many bodies in

our solar system, and determines the spacecraft motion.

Basic n – Body System

Consider system of ‘n’ particles, moving with respect to

an arbitrarily chosen inertial frame, under gravity force.

Motion of n–particle system, as described above, under

the assumption of spherical symmetry of universal

gravitation law, can be derived as follows.

Applicable Equations of Motion

* *

31

Motion of i particle: m ; 1n

i jth

i i ij ij ij ij

j ij

m mr G r

rδ δ δ

=

= = −∑� �ɺɺ

Here, ‘G’ is the universal gravitational constant and δijis the kronecker delta.

Gravity represents a self-equilibriating force system.

1j ijr=

1

Summation of above for 'n' particles: m 0 (r r )n

i i ij ji

i

r=

= = −∑� ��ɺɺ

Resulting system of equations is a set of ‘n’ 2nd order

ODEs, which can be symbolically integrated as follows.

n – Body Motion Formulation

1 1 ; Mass of 'n' Particles

n n

i i i i

i ic n

m r m r

r MM

= == = →∑ ∑

∑

� �

�

Here, c1 and c2 are arbitrary constants of integration

and the solution indicates that centre of mass is either

stationary or moves with uniform velocity.

1

1 2

1 1

0

n

i

i

n n

i i c i i c

i i

Mm

m r Mr m r c t c Mr

=

= =

= = → = + =

∑

∑ ∑� � � � � �ɺɺ ɺɺ

In order to explore the solution further, we can carry out

special types of vector and scalar products of the

differential equation, as follows.

n – Body Motion Formulation

Torque or Moment For i Particlethi i i in n n n

r m r H

d

× = →

∑ ∑ ∑ ∑

�� � ɺɺɺ

� � � � � � � �ɺ ɺ ɺɺ ɺ ɺɺ

( ) ( )

1 1 1 1

* *

3 31 1 1 1

0

n n n n

i i i i i i i i i i

i i i i

n n n ni j i j

i ij j i ij i j i

i j i jij ij

i

dH H r m r m r r m r r

dt

Gm m Gm mH r r r r r r

r r

H r

δ δ

= = = =

= = = =

= = × = × = ×

= × − = × −

= ×

∑ ∑ ∑ ∑

∑∑ ∑∑

� � � � � � � �ɺ ɺ ɺɺ ɺ ɺɺ

� � � � � � �ɺ

� � �ɺ ( )1

Constant

Conservation of Angular Momentum

n

j j i i i i

i

r r r H m r r=

= − × → = × =∑�� � � �ɺ

Similarly,

n – Body Motion Formulation

( ) ( )

( ) ( )

*

31 1 1 1

*

3,

n n n ni j

i i i i i i ij i j i

i i i j ij

n ni j

ij j i j i

m mm r r r m r G r r r

r

m mdT dVG r r r r

dt r dt

δ

δ

= = = =

⋅ = ⋅ = ⋅ −

= − ⋅ − = −

∑ ∑ ∑∑

∑∑

� � � � � � �ɺ ɺɺ ɺ ɺɺ ɺ

� � � �ɺ ɺ( ) ( )31 1

1

1 1

,

;

1Constant

2

Co

ij j i j i

i j ij

n ni j

i j i ij

n n ni j

i i i

i i j i ij

G r r r rdt r dt

m mV G E T V

r

m mE m r r G

r

δ= =

= >

= = >

= − ⋅ − = −

= − = +

= ⋅ − =

∑∑

∑∑

∑ ∑∑� �ɺ ɺ

nservation of Total Energy

Following are the gross motion parameters of an ‘n’

body system moving under pure gravitational forces.

N – Body Motion Parameters

1 2

1

Centre of Mass:

n

i i

i

m r c t c=

= +∑� � �

1

Momentum Conservation:

n

i i i

i

r m r H=

× =∑�� �ɺ

These are a set of 10 equations in ‘6n’ unknowns.

1i= 1i=

∑

2

1 1

Energy Conservation:

1

2

n n ni j

i i

i i j i ij

m mm r G E

r= = >

− = ∑ ∑∑ɺ

Practical Aspects of Space Motion

As gravitational pull of a body depends on its mass and

its proximity to another body, presence of large body

close by can be considered as a dominant effect.

E.g., as sun, the largest body in our solar system, is

closest to all planets in our system, planets’ motion areclosest to all planets in our system, planets’ motion are

determined primarily by the sun’s gravitational pull.

Similarly, as moon of Earth is closest to Earth, its orbit is

largely decided by the gravitational pull only of Earth.

This has resulted in approximate, but simplified, closed

form solutions for Earth – satellite system. 2-body

formulation is one such popular simplification.

Consider the two-body system given below.

Basics of 2-Body Formulation

2 1r R R= −� ��

As centre of mass has no acceleration, it can be treated as

the origin of an inertial frame of reference and all position

and velocity vectors can be defined as relative positions

and relative velocities, as shown above.

Basic equations of motion for a 2-body system are as given

below.

Basics of 2-Body Formulation

( )

( )

( )

( )

1 2 2 1 1 2 2 1

1 1 2 23 3

2 1 2 1

,Gm m R R Gm m R R

m R m R

R R R R

− −= − = +

− −

� � � �

� �ɺɺ ɺɺ� � � �

� �

1 1 2 22 1 1 2

2 21 2 1

2 2 1 21 1 1 2 2 2 0 1 1 2 2 0

= ; ;

; 1

1 1,

2 2

c

C C

m R m RR r R R M m m

M

m mR R r R r R R r

M M

Gm mm R R m R R H m R m R E

r

+= − = +

= + = + = + +

× + × = + − =

� �� � ��

� � � � �� � �

� � � � �ɺ ɺ ɺ ɺ�

Summary

An object, when injected in space, exhibits motion

which is governed broadly by Kepler’s laws.

Basic governing equations arise from the application of

Newton’s universal law of gravitation.Newton’s universal law of gravitation.

In a scalar sense, governing equations represent a total of

10 linearly independent equations in ‘6n’ unknowns.

Simplification are made that result in useful closed form

solutions of general n-body equations.