Orbital mechanics

Rxke · 2004-03-25 02:05:59

If it's stuff like Mars-Ceres-Moon-Earth trajectories, we'll probably have to go to a more sophisticated system.

Now wouldn't that be a great project to crunch numbers over?

You could send a light (=fairly cheap) probe on its way, launched with a low-powered (again fairly cheap) booster, after a group of enthousiasts did the numbers...

The mini-sat on auction at eBay, a while ago comes to mind, IIRC, it could be fitted with a small 'kick-motor,' though the payload was minimal... But the cost, inclusive launcher was *very low*

What about Earth-Mars-Phobos-Deimos trajectories... (Phobos and Deimos *still* are not really photographed in detail, though MarsExpress tends to do something about that, sometime, at least it was one of their options in the mission protocols...

SBird · 2004-03-25 02:17:32

BTW, look at NASA's Genesis mission for an example of these trajectories. Martin Lo did the mission planning for it. The spacecraft uses NO course corrective thrusts except for instrument pointing after hitting Lunar L1 - pretty amazing.

As for a Mars Society mission, you still have to get from LEO to lunar L1 which isn't cheap. Maybe, though if our probe were small enough, we could piggyback a ride on a geosynchronous satellite or something. It's a cool idea.

SBird · 2004-03-25 02:18:48

??? Uhhh.. why did I post duplicate messages earlier? Did the board hiccup or something?

Rxke · 2004-03-25 02:42:50

As for a Mars Society mission, you still have to get from LEO to lunar L1 which isn't cheap.

Which again causes me to dust off my favourite war-cry:

"A tug! A tug! My kingdom for a tug!"

The piggypacking idea is a nice one, but privately owned sat launches seem to be a bit adverse to that way of doing business, it would be good PR, though, sending a small probe together with their hulking mega-comm-sats into the deep (or high?)

And science missions that tend to be more positive to this approach aren't all that common for GEO, so i'd guess there's a *huge* waiting list for "hikers" ... sigh..

Rxke · 2004-03-25 06:26:53

While I'm (again) at the wild-ideas-path...

How could one get an ice-asteroid in a libration point? Any libration point...

If possible, then add an ion- thruster and set it on course to orbit Earth, as a source of future propellant, or even more wild: let it aerobreak around Mars, to start terraforming...

John Creighton · 2004-03-25 07:25:52

Wouldn't an ice asteroid be a commet? Comments have a natural thrust.

John Creighton · 2004-03-25 07:51:57

abstract from:
[http://www.gg.caltech.edu/~mwl/publicat … stract.pdf]http://www.gg.caltech.edu/~mwl....act.pdf
Paper at: [http://www.gg.caltech.edu/~mwl/publicat … ateway.pdf]http://www.gg.caltech.edu/~mwl/publicat … ateway.pdf

The Lunar L1 Gateway: Portal to the Stars and Beyond
Martin Lo, Shane Ross
Our Solar System is interconnected by a vast system of tunnels winding around the Sun generated by
Lagrange Points of all the planets and their moons. These passageways are identified by portals around
L1 and L2, the halo orbits. By passing through a halo orbit portal, one enters this ancient and colossal
labyrinth of the Sun. This natural Interplanetary Superhighway System (IPS) provides ultra-low energy
transport throughout the Earth’s Neighborhood, the region between Earth’s L1 and L2. This is enabled
a coincidence: the current energy levels of the Earth L1 and L2 Lagrange points differ from that of the
Earth-Moon by only about 50 m/s (as measured by rV). The significance of this happy coincidence
the development of space cannot be overstated. For example, this implies that lunar L1 halo orbits are
connected to halo orbits around Earth’s L1 or L2 via low energy pathways. Many of NASA’s future
observatories located around the Earth’s L1 or L2 may be built in a lunar L1 orbit and conveyed to the
destination via IPS with minimal propulsion requirements. Similarly, when the spacecraft or instruments
require servicing, they may be returned from Earth libration orbits to the Lunar L1 orbit where human
servicing may be performed. Since the lunar L1 orbit may be reached from Earth in less than a week,
infrastructure and complexity of long-term space travel is greatly mitigated. The same orbit could reach
point on the surface of the Moon within hours, thus this portal is also a perfect location for the return
human presence on the Moon. The lunar L1 orbit is also an excellent point of departure for interplanetary
flight where several lunar and Earth encounters may be added to further reduce the launch cost and
the launch period. The lunar L1 is a versatile hub for a space transportation system of the future.

Rxke · 2004-03-25 09:12:44

Wouldn't an ice asteroid be a commet? Comments have a natural thrust.

commet? Comment? Comet? Confusing post, heehee..

What do you mean with natural thrust? Their tails? or... the 'geysers'... But they're all in a different direction... Would the net thrust not be negligable? Hmmm... probly not compared to a small ion-thruster... Now I see your point.

John Creighton · 2004-03-25 12:41:59

There seems to be a lot of discussion about these minimum energy tunnels. Although they are interesting, I am still interested in more classical transfer orbit. I was thinking of next finding the plane of a more realistic orbit from the actual coordinates I found. Then I will try and compute the transit time for this orbit. I will then compute the delta v for the this orbit with the minimum d value. I will then compute the orbit for 1.5 times the minimum d value. I will then try to write code to plot the delta v and transit time to go from a single point on earth’s trajectory to each point on the trajectory of mars. Several of these curves will be plotted on the same graph. Each curve will be for different multiples of the minimum value of d. any suggestions?

Recall I used d to represent the sum of the distance from each foci of the trajectory.

Bill White · 2004-03-25 14:49:45

There seems to be a lot of discussion about these minimum energy tunnels. Although they are interesting, I am still interested in more classical transfer orbit.

I have created a new thread to discuss minimum energy transit trajectories.

SBird · 2004-03-25 14:55:08

The 0 energy trajectories are neat but the energy savings to Mars aren't overwhelming - you still have to get to Moon L1 before you can use one. Plus it's probably too slow for human transfer. Nonetheless, it's a very good way to increase cargo mission performance and unlike standard orbits, hasn't been worked out in detail yet.

If I get a chance, (unlikely for a few months) I'll try to write up an orbit simulator that would let one play around with various delta Vs.

SBird · 2004-03-25 14:56:16

PS: the orbit calculator would be for standard orbits, not the newfangled ones.

John Creighton · 2004-03-28 16:18:18

I worked out the eqution of the ellips that has one focus of the sun and the other focus given by the coordinates (f1,f2)

It is:
a1 x^2+a2 x +a3 x y + a4 y +a5 y^2 =a6
a1=(4 d^2 - 4 (f1 -s1))
a2=s k (f1 -s1) - 8 d^2 s1
a3=-8 (f1-s1) (f2-s2)
a4= 2 k (f2 - s2) - 8 d^2 s2
a5 4 d^2 -4(f2-s2)^2
a6=(||f||^2-d^2)^2+(||s||^2-d^2)2-d^4

where
d is the sum of the distance from each foci to a point on the ellips
(s1,s2) are the coordinates of the sun
(f1,f2) are the coordinates of the second foci
k=||f||^2-||s||^2-d^2
||f|| denotes the maginitude of the postion vector giving the location of the second foci

John Creighton · 2004-03-30 07:57:25

Getting the Velocity in Vector Form

Given the equation of the [http://mathworld.wolfram.com/ConicSection.html]conic section:
a1 x^2 +a2 x + a3 x y + a4 y +a5 y^2= a 6
I worked out the [http://mathworld.wolfram.com/Derivative.html]derivative
dr/d_theta by [http://mathworld.wolfram.com/ImplicitDi … ation.html]implicit differentiation:
This is relevant because the [http://mathworld.wolfram.com/Slope.html]slope is related to the derivative by:
dx/dy=((dr/d_theta) sin(theta) + r cos(theta))/((dr/d_theta) cos(theta) + r sin(theta))
The [http://mathworld.wolfram.com/Slope.html]slope gives the [http://mathworld.wolfram.com/Direction.html]direction a body is
traveling that is following the [http://mathworld.wolfram.com/ConicSection.html]conic section.
In other words it can be used to find the
instantaneous [http://mathworld.wolfram.com/VelocityVector.html]velocity vector.

The solution is:
dr/d_theta=

a6+a1 r^2 sin(2 theta) + a2 r sin(theta)- a3 r^2 cos(2 theta) –a4 r cos(theta) – a5 r^2 sin(2 theta)
------------------------------------------------------------------------------------------

a1 r (1 + cos(2 theta)) + a2 + a3 r + a4 +a5 r (1- cos 2 theta)

John Creighton · 2004-12-02 14:43:35

I remember a while back someone was trying to find the optimal orbits. In the two body problem I don’t know if you can do better then a the classical type of transfer (‘what was it called again) however in three or four body problems http://www.newmars.com/forums/viewtopic.php?t=2410]it is possible to better. The optimal solutions can be found by traveling along the space velocity manifolds. I was wondering if another possible procedure might be to break the trip down into a bunch of sub orbits thus allowing a search of many possible routes without resorting to numeric integration. I was thinking that each two body orbit could be calculated and for short distances the resulting orbit is roughly the sum of the three orbits. Sounds a little adhock I know but it should work really well in regions where one of the orbits dominate. The optimization algorithm would start with one maneuver transfers then search possible two maneuver transfers and so on. A look up table could be created for the date the desired trip time to mars and the delta V and minimum average thrust required. This look up table could be later searched for the nearest suitable transfer time.

P.S. Part of my goal here is to find this thread again because I might make attempt at flight planning algorithms before I try anY computational fluid mechanics to test http://www.newmars.com/forums/viewtopic.php?t=2504](1) OR http://www.newmars.com/forums/viewtopic.php?t=2567](2) I will need for my pipe problem or the creation of a http://www.newmars.com/forums/viewtopic … 2292]super general simulation architecture.

ftlwright · 2004-12-02 18:40:17

Here is an excel file made a couple years ago comparing the specific impulse/mass fraction/transfer time for a hohmann transfer to mars.

http://homepage.mac.com/dbjwright/FileS … .html]mars to earth spreadsheet

Be patient if you have any questions, it's been a while since I last took a look at this.

John Creighton · 2004-12-03 10:43:33

Here is an excel file made a couple years ago comparing the specific impulse/mass fraction/transfer time for a hohmann transfer to mars.
mars to earth spreadsheet
Be patient if you have any questions, it's been a while since I last took a look at this.

Looks pretty cool. I think I got most of it. All flights are hohmann transfers leaving from earth. (I forget for mars is that a conjunction or opposition class mission) . I presume Mo/Mf is the ratio of the initial mass to the final mass. I presume rp is distance from the sun at departure (paraphilian) and ra is distance from the sun at arrival (aphilinan). I assume you looked at more then just arriving at mars considering you have a plot of the paraphilian vs delta V. As far as mission planning goes I think the mass fraction and transfer time tables are the most useful. The only problem is not all launch technologies have the thrust to do a hohmann transfer. I presume the mass fraction stays the same but the flight time changes.

John Creighton · 2004-12-03 11:02:11

What ISP does Zubrin assume for the mars mission 200 s? For an ISP of 200 it looks like the mass fraction is 7/3 but this is just for the transfer and not to escape earth, circularize or to land and some of this weight is going to be the vehicle. That is if I read the graph right is mars 1.75 Au from the sun? (Maybe I will look this up later.)

John Creighton · 2004-12-03 16:21:49

On the topic of excel I am currently playing around with visual basic and excel. I would rather program in MATLAB but allot more people nave excel then MATLAB. Maybe I will try examples for both. A set of useful flight planning excel functions could be quite popular. I don't like excels graphs. Maybe I will sometime write my own graphing modal for them in JAVA or something.

ftlwright · 2004-12-03 16:35:42

Mars is ~ 1.5 AU, you can substitute this with a more accurate number if you like.

The first two tables look at transfer ellipses with Apehelion from 1 to 8 while the lower two tables look at ellipses from 1 to 3. The only reason this was done was to get a better idea of what changes in energy did to the energy requirement.

When looking at energy for departing and arrive from earth/mars systems, just used a hyperbolic trajectory for the analysis. From there you should be able to size a launch vehicle for earth departure.

FYI, the deltaV requirement for escaping the earth Sphere of Influence is about 11k m/s.

If you want to share you work w/ others (especially for these 'back of the envelope problems' just use a spreadsheet program. When your equations become more envolved use Matlab. You don't want to waste time learning a new language when the tools are alread available to you. Also, most '*.m' files can be run in Octave, so that's always an option for folks you would like to share your work with.

John Creighton · 2004-12-03 18:12:37

Learning MATLAB is much easier then trying to learn how to use Visual Basic Macros for excel. However more people have excel so it is worth my time learning how to create some macros for the spread sheets. Right now the VB runtime errors are giving me a headache and I am just experimenting with simple things.

remcook · 2004-12-04 05:10:29

John...if you want to keep it simple first, you can start with calculating some Hohmann transfer trajectories...

http://en.wikipedia.org/wiki/Hohmann_tr … sfer_orbit

John Creighton · 2004-12-04 13:59:28

John...if you want to keep it simple first, you can start with calculating some Hohmann transfer trajectories...

O.K. First a more complicated version to demonstrate the language then a mediumistic version to follow.

To create a VB macro in excel go to tools, then macros, then visual basic editor. Make sure you have the right project selected. This is only relevant if you have more then one spread sheet open. Then go to insert and select new modal.

Bellow is the VB code to calculate the departure delta V. Note that it is not necessary to convert everything to a double. You can perform mathematical operations on ranges just like you can with doubles. I just included it because I am new to visual basic and I would like to know how to convert things if I needed to. A simpler version will follow.

After copying and pasting the code in the new module. Save it. Go to debug and select CompleVBA Project. Then Go to Run and select Run Sub/User Form. Select Hohmann_test then push the Run button. You can add break points by clicking on the gray bar to the right of your code. This will stop the execution of your code so you make sure the variables are what you expect they are. To check to see what the variables are go to Debug and select add watch. In future code samples I may leave out the test function.

[code:1]Sub Hohmann_test()
   Dim AU As Range
Dim G As Range
Dim r1 As Range
Dim r2 As Range
   Dim answer As Double
Set AU = Worksheets("Sheet1").Range("B1")
   Set G = Worksheets("Sheet1").Range("B2")
   Set r1 = Worksheets("Sheet1").Range("B3")
   Set r2 = Worksheets("Sheet1").Range("B4")
   answer = Hohmann_P(r1 * AU, r2 * AU, G)
End Sub
Function Hohmann_P(r1_a, r2_a, mu_a)
   Dim r1 As Double
   Dim r2 As Double
   Dim mu As Double
   r1 = DblFrmRange(r1_a)
   r2 = DblFrmRange(r2_a)
   mu = DblFrmRange(mu_a)
   Hohmann_P = Sqr(mu / r1) * Sqr(2 * r2 / (r1 + r2) - 1)
End Function
Function DblFrmRange(a) As Double
   If StrComp(TypeName(a), "Range") Eqv 0 Then
DblFrmRange = CDbl(a.Value2)
   Else
DblFrmRange = CDbl(a)
End If
End Function[/code:1]

John Creighton · 2004-12-04 15:10:48

I ran the code but I got the wrong answer. I assumed mu was the universial gravitatial constant. Maybe it is the gravity due to the sun at r1. Maybe. Here are my numbers:

Excell Call:
[code:1]=Hohmann_P($B$3*$B$1,$B$4*$B$1,$B$2)

   A    B          C (Answer)
1    1 AU (m)    149,597,870,000 1.10288E-11
2    G (N m^2 Kg^-2)    6.672E-11
3    Earth (AU)          1
4    Mars (AU)    1.75[/code:1]

John Creighton · 2004-12-04 15:14:38

Here is the simplified visual basic Version:

[code:1]
Function Hohmann_P(r1, r2, mu)
Hohmann_P = Sqr(mu / r1) * Sqr(2 * r2 / (r1 + r2) - 1)
End Function[/code:1]

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Re: Orbital mechanics

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