Trip to Mars

1
Trip to Mars

• How do we get there?
• OAPT May 08
• By John Berrigan
• Berriganj_at_hdsb.ca

2
The Theory
How do we get from Earth to Mars?
3
The Problem

• The trip to Mars is a complicated multibody
  problem.
• The main players are
• Probe
• Earth
• Mars
• Sun
• Jupiter (not a major player but for long trips
  can move you off course)

4
The Solution

• We will change the multi-body problem down to a
  series of two body problems.
• Earth/probe
• Sun/probe
• Mars/probe
• The result gives a pretty accurate representation
  of what needs to be done.

5
How should we get there?

• Traveling in space is expensive. At present,
  depending upon the source, it costs around
  10,000 / kilogram to put into low Earth orbit,
  so fuel savings is important.
• Less fuel needed for trip less cost.
• As well, launching payloads to orbit can mean
  large launch increases. If the payload reaches a
  certain mass, a more expensive launcher is
  needed.
• The Hohmann transfer orbit is one way to minimize
  the costs.

6
Hohmann Transfer Orbit
We want to go from the inner orbit to the outer
orbit.
The red orbit is the smallest transfer orbit from
the lower orbit to the higher orbit. This is
called the Hohmann transfer orbit

• The Hohmann transfer orbit involves a low energy
  transfer. It only requires two boosts of energy
  or delta-vs to change orbits.

7
The Delta-vs
Destination orbit Transfer orbit ?v needed
Destination orbit Transfer orbit ?v needed

• ?v1 gets you into transfer orbit
• ?v2 gets you into destination orbit
• Both ?vs involve change in speed not direction
  since velocities are tangential to the orbit.

8
Larger Energies
The two orbits may actually have the same speed
at that pointbut the Direction change is the
main factor.

• You can do the transfer using a larger ?v on the
  first burn.
• This means a larger ?v is needed at the other
  side.
• The 2nd ?v both changes the magnitude AND
  direction.
• It is a faster route but more expensive due to
  more fuel.

9
The Physics of it All!!
10
What do we need to know?

• Ellipse properties
• FnetFcentripetal
• Energy conservation
• Kinetic Energy (½mv2)
• Gravitational Potential Energy(-GMm/r)
• Orbital Velocity Equation
• Relative motion
• Keplers 3rd Law

11
The Ellipse.

• Review on ellipses
• Objects orbit in ellipses.
• Central body at one of the focus points

VP Velocity at periapsis VA Velocity at
apoapsis
vp gt vA
12
The Ellipse continued
b
-a
a
- b
e is the eccentricity. Simply how oval it
is. Changes position of focus relative to the
x-int e 0, circle e lt 1 ellipse e 1
parabola e gt 1 hyperbola
13
Additional Jargon

• Periapsis is the closest point from a focus.
• Apoapsis is the farthest point from a focus.
• These names can be modified to the body being
  orbited
• Sun (helion) Perihelion and aphelion
• Earth (gee) Perigee and apogee
• Moon (lune) Perilune and apolune
• Mars (areion) Periareion and Apoareion

14
Energy Conservation

• How fast must you go to JUST escape the Earth?

ET ET ½mv2 -GMm/r 0 Therefore, v2 2GM/r For
Earth vescape 11.1 km/s
R infinity V 0 Therefore ET 0
ET EK EP
15
Relative motion

• From previous slide we found the escape velocity.
  This means at infinity, the velocity is zero
  relative to the Earth.
• If we change the frame of reference to the Sun,
  the Earth has a velocity. That means when the
  probe gets to infinity, the probe has the same
  speed as the Earth.

R infinity
16
Probe
Earth
Sun

• Even though the probe never gets an infinite
  distance away, we can argue that the probe is in
  the same orbit as the Earth (since it has the
  same speed) but it is outside of the Earths
  Gravitational influence.
• So we obviously cant get to Mars with just the
  escape speed.

17
So What do we need to do?
When we get to infinity, we need to have a
velocity in order to change orbits!!
But how much faster?

• As Buzz Light-year has famously said, we need to
  go
• To infinity and beyond!!!

18
Circular Orbits Orbital Energy

• To solve for the trajectory we need to review
  orbital energy.

ET EK EP ½mv2 ( -GMm/R)
½m(GM/R) GMm/R ½(-GMm/R) ½ EP
Fnet Fg Fc Fg mv2/R GMm/R2 v2 GM/R
This means that in a circular orbit the total
Energy is equal to one half the potential Energy
at that radius.
19
Elliptical Orbits

• Since a Circle is a type of ellipse we can modify
  the Total Energy equation
• ET -½GMm/R.
• The radius is really the semi-major axis so
• ET -½GMm/a
• Where a is the semi-major axis.

20
Elliptical Orbital Velocities

• We know energy is conserved so
• ET Ep EA
• ET EK EP
• -½GMm/a ½mv2 GMm/r
• Rearranging and solving for v we get
• v2 GM(2/r 1/a)

21
Advanced solution

• If you introduce angular momentum, R x V, at
  periapsis and apoapsis, R and V are
  perpendicular. Therefore,
• rpvp rAvA , we can then derive the equation.
• We know EA Ep. Therefore,
• ½mvA2 GMm/rA ½mvp2 GMm/rp
• Substitute for vp and simplify. After a bunch of
  math we get
• VA2 GM(2/rA 1/a)
• (this is a GREAT exercise for the stronger math
  students in the class!!)

22
What can we do now?

• We now can solve a good chunk of the problem!
• Find the velocity of the Earth and Mars by using
  Fnet Fg. (We will assume they are circular
  orbits.)
• Determine rA , rp , a of the transfer orbit.
  (An extension, find eccentricity of the orbit.)
• Determine vA and vp.
• This data can now be used to determine the ?vs
  needed for the transfer orbits.

23
The Orbit data and our results
Earth (Circular orbit) r 1.50e11 m, v 29.7
km/s Mars (Circular orbit) r 2.27e11 m, v
24.2 km/s Transfer orbit, (Elliptical orbit) rp
1.50e11 m, rA 2.25e11 m, a 1.885e11 m vp
32.6 km/s, vp 21.6 km/s
24
The Delta vs

• Therefore delta vs needed are
• ?v1 Vp VEarth 2.9 km/s
• ?v2 VA Vmars 2.6 km/s

These delta vs are the values for the two body
problem of the probe and the Sun.
25
Now to leave and arrive!!

• Now that we have figured out the transfer orbit,
  we now need to worry about how Mars and Earth
  affect the values.
• Using relative motion, we will now address the
  two body problem of the probe and Earth, as well
  as, the probe and Mars

26
Earth launch speed

• We found that ?v1 to be 2.9 km/s. Therefore the
  probe needs to travel 2.9 km/s faster than the
  Earth is traveling.

So, the probe, after launching from the Earth,
must have a velocity of 2.9 km/s when it gets to
infinity.
27
Orbit Transfer

• How fast must you launch from Earths surface to
  get into transfer orbit?

V 2.9 km/s
V ? R r
ET ET ½mv2 -GMm/r ½mvinfinity2 vlaunch
11.6 km/s
ET EK EP
Note there is small difference (400 m/s) in
launch velocity for JUST escaping and having
final velocity of 2.9 km/s.
28
Arriving at Mars

• Arriving at Mars is a little different.
• We found that Mars is traveling 2.6 km/s faster
  than the probe at the transfer point. (So Mars is
  actually catching the probe.)
• This means relative to Mars, at inifinity the
  probe is approaching Mars at a speed of 2.6 km/s.
• What ?v is needed to arrive at the planet?
• Depends!!!!!
• Do you want to land or orbit?

29
Landing on Mars

• How much should you slow down when you arrive at
  the Martians surface?

The calculation ET ET ½mv2 -GMm/r
½mvinfinity2 v 5.7 km/s
ET EK EP
So to land you need a ?v of 5.7 km/s
You are going to get this delta V
regardless.... Trick is doing it safely. Just ask
the Mars Polar Lander of 1999.. cross fingers for
tomorrows landing of the Lander's Sister, Phoenix.
30
What is the real answer?

• A quote from the FAQ from the Phoenix Lander
  site.
• Entry, Descent and Landing
• The intense period from three hours before the
  spacecraft enters Mars atmosphere until it
  reaches the ground safely is the mission phase
  called entry, descent and landing. The craft will
  hit the top of the atmosphere at a speed of 5.7
  kilometers per second (12,750 miles per hour).
  Within the next six and a half minutes, it will
  use heat-generating atmospheric friction, then a
  parachute, then firings of descent thrusters, to
  bring that velocity down to about 2.4 meters per
  second (5.4 miles per hour) just before
  touchdown.
• Not too bad for some approximations!!

31
Orbiting Mars

• To find the delta V, we first need to find the
  orbital velocity in the final orbit. Lets assume
  at an altitude of 200 km.

From earlier, v2 GM/R, so orbital velocity is
3.5 km/s.
32
Orbiting of Mars

• Now to find the velocity as probe approaches from
  infinity. If no ?v, probe does a fly by.

V ? R r
ET EK EP
The calculation ET ET ½mv2 -GMm/r
½mvinfinity2 v 5.6 km/s
To orbit you need to a ?v of 5.6 km/s - 3.5 km/s
Or 2.1 km/s
33
Quick quiz

• Lets see who is awake..
• Q What happens if you want to go into a 200 km
  circular orbit and the ?v is smaller or larger
  than the 2.1 km/s needed?
• A Since you are really taking energy away from
  the orbit when using the ?v, you are changing the
  type of conic section the final orbit will be in.

34
Orbit Energy

• If ?v 2.1 km/s orbit is a circle.

If ?v gt 2.1 km/s, final orbit energy is less.
If ?v is a little lt 2.1 km/s
Quiz 2 What ?v is too small or too large??
35
Energy

• If ?v is larger than 2.1 km/s and the Periareion
  takes us into the atmosphere.
• If ?v is smaller than 2.1 km/s and the total
  Energy relative to Mars is
• Negative ellipse (the larger the negative, the
  smaller the semi major axis, smaller the orbital
  period)
• Zero parabolic orbit (escapes)
• Positive Hyperbolic orbit (escapes)

36
Back to our problem.
37
Mom, we there yet?

• Not quite
• So far we know
• Earth ?v 11.6 km/s
• Mars ?v is
• 5.6 km/s to land
• 2.1 km/s to circular orbit
• Now we have to make sure Mars is there when we
  get there!!
• Where should Mars be when we launch?

38
When do we Launch?

• Now for Keplers law!!
• Remember T2 K R3,
• we can use this to find how long it takes to get
  to Mars and how long Mars travels in that time.
• Once again, we can modify Keplers law to any
  ellipse.
• So, T2 K R3 becomes T2 K a3 where a is the
  semi major axis.

39
Working with Keplers law

• The K value can be of any units. For ease of
  use, T is in years and a is in 1011 m.
• To find K for the sun, use Earth data.

Earth Tearth 1 year, aearth 1.5 so Ksun
1.5-3
Transfer atransfer 1.885 T2 Ksun (1.885)2 T
1.41 years
Mars aMars 2.27 T2 Ksun (2.27)2 T 1.87
years
40
Almost done..

• Transfer orbit takes 1.41 years to do a full
  orbit. So it takes 0.705 years or 8.46 months for
  half that orbit.
• How far does Mars Travel during the transit time?
• Simple ratio
• Degrees 360o __x__
• Period 1.86 0.705
• X 136o.
• So Mars travels 136o while probe heads towards
  Mars.

41
FINALLY!!!

• Since the probe arrives at Mars 180o from where
  Earth was at launch. Mars must be 180o 134o
  46o in front of the Earth at launch.

42
When can we do it again?

• Angular Velocity of Earth 360o/1 year
• Angular Velocity of Mars 360o/1.86 year.
• Difference is 166o per year
• or
• 360o change in 2.16 years or 26 months.
• Which is why we try go to Mars Every 26 months

43
Ok, what now??

• With the basics covered you can have lots of
  extensions.
• In real launches, most times the rocket puts the
  probe into a circular orbit around the Earth
  first, does a self check to see if all is well
  and then a delta v takes it to the transfer orbit.

What ?v is needed to get a Vinfinity 2.9 km/s?
44
Design a mission

• To have the arrival orbit as an ellipse.
• To land on an asteroid.
• To Orbit an asteroid.
• To the moon.
• To change orbit altitude around Earth.
• To dock to Space Station once in orbit.
• Calculate Delta V to land the shuttle
• Note Keep the objects orbits circular for ease
  of calculation. Ellipses make it harder to
  figure out where the planet is at a given time.
  (That can be another presentation.)

45
How can you mark it???

• Answers can be easily created in excel.
• Give each group data for a planet.
• Minimizes copying. But encourages discussion
  among groups
• You just check if they are right or not.
• I have a program that I get the kids to plug
  numbers into to check if they are right.

46
Multibody problem method

• Can Involve Weak Stability Boundary
• No empirical solution
• Can involve chaotic effects
• Uses MUCH less fuel
• Langrange points can be used

Golf putting analogy Two body problem ignores
little dips and valleys on the green. Power the
putt over the breaks. Multibody problem can take
the dips into account, putt more slowly, ball
JUST drops into the cup.
47
Lagrange points
Earth Sun Lagrange points or libation points
48
Lagrange points
Gravitational topographical force map
49
Phoenix Landerfrom April 25th
50
May 23rd
51
Phoenix's Trajectory
52
Phoenix's Landing
What time will Phoenix land on Mars? What time
will the first signal be received from
Phoenix? Phoenix will land at approximately
436pm Pacific Daylight Time (736pm Eastern
Daylight Time). We hope to receive the first
signal from the lander approximately 17 minutes
later at 453pm PDT (753pm EDT).
Discovery channel has live feed at 700 pm on
Sunday.Live NASA coverage starts at 445..go to
their web site
53
How can we get to the moon?
54
Resources

• Fundamentals of Astrodynamics by Bate, Mueller
  and White
• Fly Me to the Moon An Insider's Guide to the New
  Science of Space Travel by Edward Belbruno
• Orbiter Spaceflight simulator by Martin
  Schweiger. A FREE program. A STEEP learning
  curve but fun. NOT a game!!
• SpaceX.com some cool goings on.

55
Some Orbit misconceptionsOrbital Period

• Period is independent of eccentricity.
• Since T2 K a3, the only factor is the
  semi-major axis. How oval it is, is irrelevant.

56
Orbital Velocity

• Velocity is independent of eccentricity.
• Since v2 GM(2/r 1/a), this shows that the
  velocity of the object is only a function radius
  if the semi major axis is the same.

57
Orbit Change

• ?v towards the ground does not lower the
  satellite.
• It would put it in a higher orbit since the final
  velocity would be higher then the start so the
  overall energy is higher (less negative) which
  means larger semimajor axis since ET -GMm/a.

58
Docking

• If you are behind an object, you slow down to
  dock with it.
• Slightly Counter intuitive.
• But, if you speed up significantly to try docking
  you would actually drift away.
• Faster speed. Larger semi-major axis. Higher you
  go, slower your speed, object gets farther in
  front.

59
Conclusion

• Robert A. Heinlein, "Once you make it to orbit,
  you're half-way to anywhere."

60
Space tidbits

• Spacex 2nd launch shut down
• Pressure was too low on first attempt so
  scrubbed..warmed up fuel..launched 1 hour later
• Slight Bias..Lets hope SpaceX is successful Next
  launch June 24th..hopefully
• Off topic
• Teslamotors
• Bigelow.. Two orbiting stations
• Virigin Galactic..
• First two space craft are
• VSS Enterprise
• VSS Voyager
• Google lunar X-prize
• 14 teams now..30 million dollar prize
• Mars Science Laboratory (MSL)
• Launch Sept 2009, may be last Mars probe for a
  while
• Lunar reconnaissance Orbiter
• November launch

61
MSL. It is BIG
62
(No Transcript)