Interplanetary Travel

Interplanetary Travel Concept

Patched Conic HypothesisPatched Conic Hypothesis

Departure & Arrival Manoeuvres

Interplanetary travel is concerned with motion of

manmade objects when these travel through outer space,

passing many planets in the process.

Such motions need clear understanding of the changing

Interplanetary Travel Concept

nature of forces as well as strength of the gravitational

field of the planets involved.

In reality, we need to model such motions using the

general n-body equations, but as we have solutions only

for the 2-body and restricted 3-body problems, such an

approach is not feasible.

In practice, it is found that good solutions to inter-

planetary trajectories are possible by considering the

complete trajectory as a sequence of multiple 2-body

segments, ‘joined’ together at a suitable common point.

motion of a spacecraft earth

Interplanetary Travel Concept

Thus, the motion of a spacecraft starting from earth and

going to moon, mars, jupiter etc. can be captured by

solving successively, a number of 2-body problems.

Concept of ‘patched conics’, for a smooth transition

between segments, is used to synthesize total trajectory.

Sphere of activity (SOI) represents the common point.

Patched conic hypothesis can be investigated as follows.

A spacecraft, when escaping from a planet on a

hyperbolic path, reaches the edge of SOI of that planet.

At this point, the spacecraft is assumed to become

Patched Conic Hypothesis

heliocentric, which represents a point of patching

between the two trajectories (Departure).

Next, when the spacecraft reaches the SOI of the target

planet, it again becomes planeto-centric and this point

now represents the second patch (Arrival).

At each patch point, velocity is the ‘patch’ parameter.

Departure Concept

Departure from a planet needs a change of reference

frame at the boundary of the SOI. Consider the

schematic of departure as given below.

In escaping from earth, ‘V∞∞∞∞’ is the scalar velocity at

infinity, (or the edge of SOI). Assuming circular parking

orbit & outgoing asymptote aligned with earth’s orbital

vector, we get the patch relations as follows.

Patched Conic Relations

20; ; ;

2t

R RR R r V V v V v a ⊕

⊕ ⊕ ⊕ ∞

+= + = + = + =� � � � �� � �

0

12

2 2

0

0

2 2

0 0

0 0 0

10

2

2 1 1; ;

2 2

2 2; ; ;

1; 1 ; cos

2

t

Helio

t

Helio

V v vR a r

v v v V V V vr r r

ra e

a e

µ µ µε

µ µ µ

µθ ψ

ε

⊕ ⊕ ⊕ ∞

⊕∞

⊕ ⊕ ⊕∞ ∞ ⊕ ∞

−⊕

= − = − =

= + ≈ − ∆ = + −

−= − = − = =

⊙ ⊙

A 3-d view of the possible departure trajectories is given

below, with locus of all possible points being a cone.

Departure Solution Features

Patch Condition Example

A spacecraft is required to escape from surface of non-

rotating earth so that its V∞∞∞∞ = 2700 m/s. Determine the

nominal V0. Also, what would be the new V∞∞∞∞ if the V0 is

higher by 10%. (µEarth = 3.986x1014, RE = 6378 km).

142 3.986 10× × 14

6

2 2

0 min

2

0 min 0 0

2

0 min 0

2 3.986 1011,180 /

6.378 10

11,570 /

15.18

esc parabola

no al esc parabola

no alE

no al

V m s

V V V m s

V dV dVdV dr

V rV r V V V

µ

−

− − ∞

−∞

∞ ∞ ∞ −

× ×= =

×

= + =

= ⋅ = ⋅ = ×

Arrival at a Planet

Once a spacecraft is put on a heliocentric Hohmann

transfer ellipse through departure manoeuvre, it will

arrive at the destination planet on this ellipse.

The first point of contact with the planet is the SOI of

that planet at which point, the spacecraft comes underthat planet at which point, the spacecraft comes under

the influence of the gravity of the target planet.

From this point onwards, planeto-centric analysis will be

required to determine whether the spacecraft will have

a flyby, will form an orbit or will impact its surface.

Usually, our interest is for a flyby or for an orbit (called

capture), so we first determine the impact conditions.

Conditions for Impact on Planet

To derive conditions for impact, consider the case when

the spacecraft just grazes planet’s surface, as below.

Impact Parameter or

Stand-off Distance

b →

Impact Parameter Solution

Following relations provide the solution for the impact

parameter ‘b’.

cos ; cosp p

bh V r V b V r

rφ φ∞ ∞ ∞

∞

= = = =

2 22 2

22 2

2

2

2 2

2; ; 1

p

p

p p

escesc p esc p

p

r

V VV V

r r

VV V V V b r

r V

µ µ

µ

∞

∞∞

∞

∞

− = → = +

= = + = +

Conditions for Impact on Planet

Impact parameter ‘b’ is the minimum distance that is

permitted for no impact. We can now arrive at

conditions with respect to an approach distance ‘d’.

Impact Condition Formulation

Following relations provide conditions for impact.

cos ; there will be a flyby

there will be surface graze;

there will be an impact

lim ; lim 0;

SOId r d b

d b

d b

b r b

φ= > →

= →

< →

= =

The collision (or capture) cross-section is shown below.

0lim ; lim 0;p

V Vb r b

∞ ∞→∞ →= =

Impact Condition Example

An approaching spacecraft reaches the SOI of venus with

V∞ = 2700 m/s and φ = -85o. Determine whether or not

the spacecraft will impact. (µVenus = 3.240x1014, rSOI-

Venus=0.00411AU, 1AU = 1.497x1011 m, rVenus= 6052 km).

( )

14

6

26 7

2

7

2 3.248 1010,360 /

6.052 10

103606.052 10 1 2.40 10

2700

0.00411 cos 85 0.000358 5.359 10

It will be a flyby.

esc

o

V m s

b m

d AU m

d b

× ×= =

×

= × + = ×

= × − = = ×

> →

Impact Condition Example

(b) In case it does not impact, calculate the minimum

distance from the planet surface.

11 2

2

cos 1.45 10 /SOI

h V d r V m sφ

ε

∞ ∞= = = ×

2 26 2 2

7 7

6

23.645 10 / ; 1 1.564

2

4.455 10 ; ( 1) 2.513 102

1.907 10 19,070

venus

venusp

planet

V hm s e

a m r a e m

h m km

εε

µ

µ

ε

∞= = × = + =

= − = − × = − − = ×

= × =

Summary

Interplanetary travel model is essentially an extension

of the 2-body conic solution, by ‘patching’ different

segments of the trajectory.

Departure hyperbola demonstrates the constraints thatDeparture hyperbola demonstrates the constraints that

are implicitly applied on interplanetary mission starting

point.

Arrival manoeuvre is the counterpart of departure, with

patch condition applied at the planet SOI.