Interplanetary Earth to Mars

7/29/2019 Interplanetary Earth to Mars

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Interplanetary

Transfer


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A Simplified Example

A transfer from an orbit around a inner planet tothe orbit around an outer planet.

For example, to calculate the Dv required tolaunch a spacecraft from an orbit around the Earth

(let us take LEO of 300 km altitude) to an orbit

around the Mars.


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Hohmann Transfer (from earth to Mars)

seV

sp eV t

sp mV tsm

V


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Departure from Earth


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Interplanetary Transfer (Departure)

Here = The velocity of the space probe

relative to the sun at the time of escaping earth (ie.)

velocity of the space probe on the transfer ellipse at

the departure point D (perihelion in the heliocentric

orbit).

= the circular orbital speed of the planet 1

(here, the Earth) relative to the sun.

Here | | always > | |.

sp

e

V t

seV

sp eV tseV


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Thus, by subtracting the known value of velocity

of the planet relative to the sun, one obtains the

hyperbolic excess speed on the earth escape

hyperbola.

Therefore, the required Dv at D could be

2

1 2 1

21

sp se

D e

ep

e

Rv V t V

R R R

V t V

D


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2.946 km/s --- Is it the launch speed of the space

probe at the orbit around the Earth?

8

1 2

11 3 2

11

1 8

29.783 / 0.09891

1 1.496 10 , 1.5241.327 10 /

2 1.5241.327 10 11.496 10 1.524 1

2.946 /

sun

km s

R AU km R AUkm s

AUvAU AU

km s

D


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NO! --- It is the velocity of the space probe after

escape from Earth (after launching from the earth &

achieving a great distance from the earth). This

speed is called excess speed (not the escape speed).

Therefore, we have to find out what could be the

velocity of the space probe relative to earth at thetime of launch, , in order to achieve the

excess speed of 2.946 km/s at the point of exiting

the earths sphere of influence.

2.946 /eV t km s

ep LV t


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Conservation of specific energy to calculate the

launch speed, required to achieve excess

speed of 2.946 km/s. ep LV t

2 2

2

2

3 2

1 1

2 2

10

2

2

398600.4418 / , 6378 300

11.3 /

ep epe eL ep

L

e

ep eL ep

L

ep

e L

ep

L

V t V t

r t r

Vr

V t Vr t

km s r t km

V t km s


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Therefore, at this point, the Dv applied by the

thrusters would be 11.3 7.726 = 3.574 km/s.

11.3 /ep

LV t km s

6678 km

7.726 /km s


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Thus after applying the velocity change of 3.574km/s to the original circular velocity of 7.726 km/s,

the spacecraft (or space probe) is travelling with

velocity 2.946 km/s relative to earth, or velocity of

32.7 km/s relative to the Sun.


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Arrival at Mars


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Arrival at Mars

Here = The velocity of the space probe

relative to the sun at the time of approaching mars

(ie.) velocity of the space probe on the transfer

ellipse at the arrival point A (aphelion in the

heliocentric orbit).

= the circular orbital speed of the planet 2

(here, the Mars) relative to the sun.

Here | | > | |.

sp mV t

smV

smV sp mV t


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Arrival at Mars

Therefore, the required Dv at A could be

Plugging all the values in the above equation,

2.650 km/s.

1

2 2 1

21

sm sp

A m

mpm

Rv V V t

R R R

V t V

D

mV t


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Arrival at Mars

Now the space probe is approaching Mars on a

hyperbolic trajectory from an infinite distance with

velocity of 2.650 km/s relative to Mars.


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Arrival at Mars

Now, we have to find out what could be the

velocity of the space probe relative to Mars at the

time of reaching the orbit around the Mars ( ). mp mV t

22mp marsm mmp

m

V t V t r t

3 2

42,823.3 /3397

mars km sRadius of Mars km


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Arrival at Mars

Therefore, at this point, the Dv applied by the

thrusters to slow down to orbit around Mars would

be km/s.

mp mV t

2.650 /km s

mp m oMV t V


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Reference(s)

Howard D. Curtis, Orbital Mechanics forEngineering Students, Elsevier Publications,

2005.

http://web.mit.edu/12.000/www/finalprese

ntation/traj/hohman.html

Youtube videos Design of Interplanetary

Missions using Patched Conics (3 parts).
http://web.mit.edu/12.000/www/finalpresentation/traj/hohman.htmlhttp://web.mit.edu/12.000/www/finalpresentation/traj/hohman.htmlhttp://web.mit.edu/12.000/www/finalpresentation/traj/hohman.htmlhttp://web.mit.edu/12.000/www/finalpresentation/traj/hohman.html

Documents

Interplanetary Earth to Mars