Classical Mechanics final project, transfer orbits for selected planets using the Euler-Lagrangian

1. PATH TO PLANETS
Abilene Christian University
Roy Salinas

2. PROJECT OVERVIEW
Traveling to other planets is an evolutionary necessity. Path to Planets
(P2P) is aimed to address this.
Staying stagnant on Earth is due to lack of motivation, lack of cooperation between
space agencies, and lack of ability from current technology. However, plans to
expand beyond Earth can always be made.

3. PROBLEM
What needs to get tackled:
• Earth orbit relative to
other planets
• Best windows to launch
from Earth for different
planets
• The changes of orbit
• Time estimates for each
path

4. PLANET INFORMATION
▪ Controlled variables: These are the things that are kept the same throughout your experiments.
▪ Independent variable: The one variable that you purposely change and test.
▪ Dependent variable: The measure of change observed because of the independent variable. It is
important to decide how you are going to measure the change.
PLANET MASS PERIOD OF ROTATION DISTANCE FROM THE SUN
Mercury 3.285 e 23 kg 88 days 57.91 e 6 km
Venus 4.867 e 24 kg 225 days 108.2 e 6 km
Earth 5.97 e 24 kg 1 year 149.6 e 6 km
Mars 6.39 e 23 kg 687 days 227.9 e 6 km
Jupiter 1.898 e 27 kg 12 years 778.5 e 6 km
Saturn 5.683 e 26 kg 29 years 1.434 e 9 km
Uranus 8.681 e 25 kg 84 years 2.871 e 9 km
Neptune 1.024 e 26 kg 165 years 4.495 e 9 km

5. METHODS
• Methods to address this problem include utilizing the Lagrangian to
demonstrate simple paths off earth.
• The blue line represents
the path from
“observing” Earth from a
fixed reference frame off
of earth
• The orange path
represents the path if we
adjust the z-axis to pass
through the first point
on earth

6. GOALS
• Calculate the path to each planet from Earth
• Find best window from Earth
• Enlighten others about space travel
• Show others it is feasible
• Eventually one day leave Earth

7. TIME LINE
• 11.27.2018
• Initial presentation
• Submit proposal
• 12.04.2018
• Present progress
• Show plots and calculations
• 12.11.2018
• Present results of P2P
• Pack bags to leave Earth

8. P2P UPDATE
Abilene Christian University
Roy Salinas

9. EARTH TO MARS
• Need Mars to be 33 degrees from Earths semi-major axis when it is at Perigee
• Mars has an orbital period of ~687 days ( 1.882 Earth-years)
• The period of the Hohmann transfer is ~ 560 days but we only need half the
period so the actual transfer takes ~280 days

10. EARTH TO MARS
▪ There were some assumptions made to calculate these plots and
values
▪ The degree of inclination for mars and earth ( 1.850 degrees and 1.57
degrees) is small enough it can be negligible for now
▪ Made the assumption that the planets are moving at constant speed.
This is not true from Keplers second law. (A line that connects a
planet to the sun sweeps out equal areas in equal times. )
▪ Did not consider how much fuel it would consume nor energy

11. EARTH TO MARS
Escape velocity to leave Earth

12. EARTH TO MARS
• Earth is almost a circle but
unfortunately it is not :(
• Begin by starting at the apogee of the
transfer orbit
• From there work through the math
and find the thrust ratio and the
eccentricity of the transfer

13. EARTH TO MARS

14. EARTH TO MARS

15. EARTH TO MARS

16. EARTH TO MARS
▪ Things to do:
 Consider energy efficiency for Hohmann transfers to multiple planets
 Look into sling shot maneuvers v Hohmann transfers and show energy
efficiencies
 Hopefully plot the trajectories
 Use the Euler - Lagrange in polar coordinates to verify answers

17. PATH TO PLANETS
Abilene Christian University
Roy Salinas

18. PROJECT OVERVIEW
Traveling to other planets is an evolutionary necessity. Path to Planets
(P2P) is aimed to address this.
Staying stagnant on Earth is due to lack of motivation, lack of cooperation between
space agencies, and lack of ability from current technology. However, plans to
expand beyond Earth can always be made.

19. PROBLEM
What needs to get tackled:
• Earth orbit relative to
other planets
• Best windows to launch
from Earth for different
planets
• The changes of orbit
• Time estimates for each
path

20. KEPLER ORBITS
• Utilize the Kepler orbits to find the paths around each
planet
• Find the relative location of each planet to Earth
• Calculate the best position to leave earth and best time
based on orbits
• Find the change of orbits needed

21. KEPLER ORBITS

22. RESULTS OF P2P
• I have added Venus and Jupiter to the transfer
orbits
• The Hohmann transfer orbit requires the least
amount of energy
• I included the angle of inclination in respect to the
Suns equator
•

23. EARTH TO MARS

24. EARTH TO MARS

25. RESULTS OF P2P

26. RESULTS OF P2P

27. VENUS
• When Earth launches Venus has to be ~ 52
degrees to the left of Earth when Earth is at
its perigee
• Earth will be at its perigee on January 2
• Eccentricity of transfer is ~.4014
• Thrust ratio to enter transfer is ~ .83708
• Thrust ratio to enter Venus orbit is ~1.2911
• Period of transfer is ~145 days
• Inclination of 3.86 degrees to Suns equator

28. MARS
• Orbit of Mars is ~687 days (1.882 Earth
years)
• The period of the transfer orbit is ~280
days (.767 Earth years)
• Mars will have to be 33 degrees to the
right of Earth when it is at perigee
• Inclination of 5.65 degrees to Suns
equator

29. MARS
• Earth is almost a circle but
unfortunately it is not :(
• Begin by starting at the apogee of the
transfer orbit
• Eccentricity of transfer orbit is ~
.25769
• Thrust ratio to enter transfer orbit is ~
1.11226
• Thrust ratio to leave transfer and
enter mars orbit is ~.9602

30. MARS
• Calculated velocities for transfer
and orbits
• Mars has a v ~24.130 km/s
• Earth has v ~ 29.747 km/s
• Comparing the velocities to before
and after entering transfer from
Earths apogee I have a %3 error in
velocity calculations
• Comparing the velocities from
leaving transfer orbit to mars orbit
I have ~%15 error

31. JUPITER
• When Earth is at perigee Jupiter has to be 157
degrees from Earth
• Jupiter has orbital period of 4332.59 days
(11.862 Earth – years)
• Eccentricity of transfer is ~.694729
• Thrust ratio to enter transfer orbit is ~1.29107
• Thrust ratio to enter Jupiter orbit is ~1.05026
• Jupiter has an inclination of 6.09 degrees to the
Suns equator

32. ORBITAL PLANE CHANGES
• Each planet had a certain inclination and I
decided to base mine off of the inclination
of the Suns equator
• I assumed the orbits were “circular” since
the eccentricities of each planet was less
than .1
• The delta-v is velocity change to achieve
orbital plane change
• Delta-vV =2.359 km/s
• Delta-vM = 2.378 km/s
• Delta-vJ = 1.387 km/s