Orbital mechanics calculations

[pattersoncr]

Orbital mechanics calculations
I’m trying to analyze transfer trajectories between a station in LEO and one in lunar orbit.
For a given LEO station orbit, there are two launch opportunities per month (when the moon intersects the plan of the station’s orbit). At either of these points, an off plane transfer can be initiated with no ΔV required out of the plant of the station’s orbit (the TLI burn is purely prograde). For a transfer at any other time, some out of plane ΔV is required (since the plan of the current orbit does not intersect the moon). Is the angle associated with this value the same thing as the term EIn used in IMFD? I’d like to see how “wide” the launch windows are, in other words, how long before/after the time when the moon intersects the station’s orbit can you launch from the station without incurring a large ΔV penalty. I have been able to calculate the required prograde ΔV for an Earth to moon transfer (ignoring the moon’s gravity) but I don’t know how to calculate the plane change ΔV. Here’s my question:

How do you calculate the plane change ΔV associated with an off plane transfer in which the target position (upon arrival) does not coincide with the current orbital plane? How is EIn calculated in IMFD?

 

[Andy44]

Actually, I think you have the initial assumption wrong.

For any satellite in LEO, there are two lunar transfer windows every revolution, which for a LEO is about every 90 minutes.

Now, if you are trying to get into a particular lunar orbit plane when you arrive at the moon, there are only two windows per month.

For the special case in which the earth lies in the target lunar orbit plane (lunar equatorial orbit), this does not apply.

---------- Post added at 12:21 AM ---------- Previous post was at 12:16 AM ----------

BTW, the same logic applies both ways. For (almost) any lunar orbit, there are two opportunites to transfer to LEO every rev, but to reach a particular LEO orbit plane there are only 2 windows per month for a low delta-V transfer. Special case exception is when the target LEO plane is also the moon's orbit plane.

 

[tblaxland]

Actually, I think you have the initial assumption wrong.

For any satellite in LEO, there are two lunar transfer windows every revolution, which for a LEO is about every 90 minutes.

No he is correct, two windows per month for a purely prograde TLI. Any other time of the month requires a plane change element to the TLI burn.

How do you calculate the plane change ΔV associated with an off plane transfer in which the target position (upon arrival) does not coincide with the current orbital plane? How is EIn calculated in IMFD?

The normalised cross product of the Moon's position at intercept with the TLI burn position will tell you the transfer plane. You know the post-TLI velocity (if I understood your previous thread correctly) and the direction of that vector can be determined by the cross product of the transfer plane normal with the normalised radius vector at TLI (assuming an instantaneous burn and that you are burning at the optimum time). The vector difference between the post-TLI vector and the pre-TLI velocity vector will give you the total required dV.

 

[pattersoncr]

WHOOOSH!
That was the sound of most of that going over my head.
Admittedly, my advanced math skills have atrophied quite a bit in the 10+ years since I was a full time student. I vaguely remeber what a cross product is (as opossed to a dot product) but not how to work them. I guess I have some re-learing to do!

 

[Andy44]

[QUOTE]No he is correct, two windows per month for a purely prograde TLI. Any other time of the month requires a plane change element to the TLI burn.
[/QUOTE]

I fail. You are correct. Serves me right for writing stuff like that past my bedtime.

 

[pattersoncr]

I'm doing some dusting off of old knowledge as well as digging through this amazing resource:

NASA SP 33 Orbital Flight Handbook

I also downloaded SP 34 Lunar flight handbook

(Search for "SP 33" & "SP 34" at
http://ntrs.nasa.gov)

Hopefully I'll eventually end up with an excel spreadsheet that shows required dV for a lunar transfer for any LEO throughout the month.

 

[pattersoncr]

I dug my college calculus book out and reviewed vector cross products.

A = a1i + a2j + a3k

B = b1i + b2j + b3k

A x B = (a1*b3 - a3*b2)i - (a1*b3 - a3*b1)j + (a1*b2 - a2*b1)k


But how do I get the moon's position and the spacecraft position expressed in the same coordinate frame?

actually, maybe what I need to know is how to convert orbital elements to state vectors? Does anyone know of a source for equations expressing this conversion that could be plugged into excel?

Where can I fine the moon's orbital elements expressed in earth-equatorial frame?

 

[tblaxland]

But how do I get the moon's position and the spacecraft position expressed in the same coordinate frame?

How's your C++? You can get them directly through the Orbiter API.

actually, maybe what I need to know is how to convert orbital elements to state vectors? Does anyone know of a source for equations expressing this conversion that could be plugged into excel?

Appendix C of the Orbiter manual contains the reverse conversion (ie, state vectors -> orbital elements). The hard part is calculating the eccentric anomaly from the mean anomaly since there is no closed form solution. I wrote a simple solution for that using Newton's method in [ame="http://www.orbithangar.com/searchid.php?ID=3825"]KOST[/ame].

Where can I fine the moon's orbital elements expressed in earth-equatorial frame?

I don't know of any. The Object Info dialog shows it in ecliptic frame.

Maybe I have just found an excuse to update [ame="http://www.orbithangar.com/searchid.php?ID=3824"]State Vector MFD[/ame] to display state vectors for objects other than the focus vessel...

 

[Andy44]

Appendix C of the Orbiter manual contains the reverse conversion (ie, state vectors -> orbital elements). The hard part is calculating the eccentric anomaly from the mean anomaly since there is no closed form solution.

Not true, since in the conversion of a state vector to an el set there is no calculation of mean anomaly and thus no need to fuss with converting it to true anomaly. Once you calculate the eccentricity vector in 3-space, true anomaly is the angle between e-hat and R, found by dot product.

 

[tblaxland]

Not true, since in the conversion of a state vector to an el set there is no calculation of mean anomaly and thus no need to fuss with converting it to true anomaly. Once you calculate the eccentricity vector in 3-space, true anomaly is the angle between e-hat and R, found by dot product.

Yes, but to calculate the eccentricity vector, you need the state vectors. The question was: how to calculate the state vectors, given the orbital elements. The Object Info dialog in Orbiter will only tell you the mean anomaly, from which you need to calculate the eccentric anomaly (by some root finding algorithm) in order to calculate the true anomaly - unless you can suggest a way to calculate the eccentricity vector directly from the mean anomaly.

Where can I fine the moon's orbital elements expressed in earth-equatorial frame?

It just occurred to me that Orbit MFD will tell you that information, including true anomaly, so you don't have to calculate it yourself. Just set REF to Earth, TGT to Moon and FRM to EQU.

 

[pattersoncr]

It just occurred to me that Orbit MFD will tell you that information, including true anomaly, so you don't have to calculate it yourself. Just set REF to Earth, TGT to Moon and FRM to EQU.

Ahh, that's perfect!
The orbit MFD alaong with my recent discovery that excel will multiply two matricies ought to simplify things quite a bit.

 

[mjessick]

www.google.com
state vector from orbital elements

This one looks good:
http://www.cdeagle.com/omnum/pdf/demosvoe.pdf

 

[tblaxland]

my recent discovery that excel will multiply two matricies ought to simplify things quite a bit.

I never knew that, but a quick search of the help revealed how. Now we've both learnt something from this :speakcool:.

 

[Andy44]

I used Excel for the better part of ten years before I finally figured that out.

 

[tblaxland]

Maybe I have just found an excuse to update State Vector MFD to display state vectors for objects other than the focus vessel...

I have now updated State Vector MFD, if you are interested: [ame="http://www.orbithangar.com/searchid.php?ID=3824"]State Vector MFD V0.9.3[/ame]

Updated documentation on OrbiterWiki: http://www.orbiterwiki.org/wiki/State_Vector_MFD

 

[flaugher]

Uh, instead of going through all of this can you not just get to the Moon first then do a fairly economical trajectory change from a high altitude? The Moon doesn't have THAT much gravity. It doesn't seem that I use up that much fuel lining up with Brighton, and I launch from all sorts of wierd times.

 

[pattersoncr]

tblaxland, thanks so much for your help so far. I'm getting closer...

The normalised cross product of the Moon's position at intercept with the TLI burn position will tell you the transfer plane. You know the post-TLI velocity (if I understood your previous thread correctly) and the direction of that vector can be determined by the cross product of the transfer plane normal with the normalised radius vector at TLI (assuming an instantaneous burn and that you are burning at the optimum time). The vector difference between the post-TLI vector and the pre-TLI velocity vector will give you the total required dV.

Thanks to NASA SP 33, I managed to get the moons position at intercept and TLI position both transformed into an earth-equatorial frame.
I made the assumption the TLI would occur 180deg away from the moon's position at intercept (in the plane of the parking orbit). I know this would be true for a hohmann transfer but I'm more interested in faster transfers. I figure 180deg out from the intercept ought to be pretty close. Does anyone know how valid this assumption is or how to figure out a better value for anomaly at TLI?
I have the normalized cross product of the moon's position at intercept and TLI position, but what does this tell me? Conceptually what does this tell me? I think the resultant vector is a unit vector that point normal to the plane of the transfer orbit.
Am I correct in thinking this?
If so, couldn't I just take the relative inclination between the parking orbit and the tranfer orbit and calculate plane change dV using the formula:
dV = (2*h/r) * sin(θ/2)with theta being the relative inclination? If the plane change is incorperated with TLI, h should probably be calculated using the average of pre & post TLI velocity.
If the above formula works, how do I compute the relative inclination?

---------- Post added at 10:17 AM ---------- Previous post was at 09:57 AM ----------

Uh, instead of going through all of this can you not just get to the Moon first then do a fairly economical trajectory change from a high altitude? The Moon doesn't have THAT much gravity. It doesn't seem that I use up that much fuel lining up with Brighton, and I launch from all sorts of wierd times.

flaugher,
If you're launching from the ground, you can find an efficient launch window once or twice per day. I'm talking about departing from a station already in LEO (think 2001). If you start out already in orbit, your launch window are much more limited. I'm trying to imagine how an established space infrastructure, with stations in LEO, stations in low lunar orbit and frequent flights between them, would work (again think 2001 or Greg Burch's LSTS add-ons). How often can you launch from the station to the moon? How far away from the optimum launch time can you launch without requiring a much higher dV? Is there a way to select the orbit of the station that maximizes the launch opportunities? Those are the questions I'm trying to answer.
Besides, I'm laerning/relearning a lot with this little project.

 

[tblaxland]

I made the assumption the TLI would occur 180deg away from the moon's position at intercept (in the plane of the parking orbit). I know this would be true for a hohmann transfer but I'm more interested in faster transfers. I figure 180deg out from the intercept ought to be pretty close. Does anyone know how valid this assumption is or how to figure out a better value for anomaly at TLI?

In the calculations you were doing before, the true anomaly at intercept tells you the angle between the Moon's position at intercept and the TLI position.

I have the normalized cross product of the moon's position at intercept and TLI position, but what does this tell me? Conceptually what does this tell me? I think the resultant vector is a unit vector that point normal to the plane of the transfer orbit.
Am I correct in thinking this?

Yes, except that if the two vectors are anti-parallel, the cross product will be zero and the transfer plane will be undefined. In that specific case, you can just set the transfer plane to be co-incident with the LEO parking orbit plane.

If so, couldn't I just take the relative inclination between the parking orbit and the tranfer orbit and calculate plane change dV using the formula:
dV = (2*h/r) * sin(θ/2)with theta being the relative inclination? If the plane change is incorperated with TLI, h should probably be calculated using the average of pre & post TLI velocity.

I'm not sure if taking the average h would work or not. It should at least give a reasonable approximation for small theta anyway.

If the above formula works, how do I compute the relative inclination?

Take the inverse cosine of the dot product of the transfer normal and the LEO parking orbit normal.

 

[pattersoncr]

The vector math for computing plane change ΔV was making my brain hurt, so as an interim step, I solved for relative inclination between the parking orbit plane and the transfer plane. I then, computed this for several throughout the course of a month (gotta’ love “fill right”). This should give an indication of the shape of the required ΔV vs. launch time (Rinc α plane change ΔV) and show the “steepness” of the launch windows. As expected, the graph shows two launch windows per month. I’m somewhat disappointed as to the “narrowness” of the launch windows (although since I don’t know the ΔV associated with the Rinc angle, I can’t really define where the launch windows end).



There is however, a problem with my math somewhere.
I used this transformation from NASA SP 33 to convert from in plane values for moon & TLI positions:




I used excel’s matrix multiplication function to find the moon’s position at intercept and TLI position with respect to the earth’s orbital plane (two similar transformations are used, one with moon orbit parameters, the other with parking orbit parameters).
As a second check, I computed the length of the moon’s radius vector
=SQRT(X2 + Y2 + Z2)
This, however did not equal the same calculation before the transformation. The error is present with both transformations. I suspect I’ve gooned up the transformations somehow but I can’t figure it out.

 

[pattersoncr]

My attempt at orbital mechanics calculations using excel is now kicking my butt.
I've attached the most recent iteration of my spreadsheet in the hopes that someone can help me with this.
I can't find anything wrong with the transformation matrices I'm using but unless I set the ω (Argument of Periapsis) for both the parking orbit and the moon's orbit to zero, I get errors in the transformation. Specifically, when I transform the moon's position into the ecliptic frame, the moon's radius vector (i.e. sqrt[x2 + y2 + z2]) does not equal the radius vector I had in the frame of the moon's orbit. A similar problem exists with the parking orbit transformation. This can bee seen by comparing rows 51 vs. 55 and rows 67 vs. 71. I graphed both of these errors in the "Charts" tab. At the moment, I'm assuming that both the moon's orbit and parking orbit are circular. This means that ω is zero (which eliminates the errors) but in the future, I'd like to take the moon's eccentricity into account.
More importantly, when I graph TLI ΔV vs departure MJD, the curve looks nothing like the reference data I compiled with IMFD. (I set up an off plane transfer with IMFD and wrote down the TLI ΔV for several departure times. For each time, I ensured that TLI occurred at the optimum point in the parking orbit by finding a local minimum in ΔV.) I would expect the actual values for ΔV to differ from IMFD based on the assumptions my calculation makes (see below) but if I'm doing things right, the two graphs ought to have the same shape. They are not even close.
My spreadsheet makes the following assumptions:

• The effect of the moons gravity is ignored, I calculated the SMa of a transfer ellipse that would place the vessel at the moons orbital radius at the desired flight time.
• Both the parking orbit and moon's orbit are assumed to be circular.
• TLI occurs at the optimum position along the parking orbit. Twice per month this should yield an off plane transfer with a purely prograde TLI. I calculated the ΔV for this transfer at 3159m/s. IMFD yields 3158m/s so I'm sure I'm doing that part of the math right.
• TLI burn is assumed to be instantaneous.
• Both orbits are assumed constant (no precession or nodal regression, etc.)

I've been looking at this thing for a week and now do not know how to proceed. Can one of you math whizzes take a look this and help me figure out where I'm going wrong?