Aligning orbital planes

[Jairo]

I was playing with orbiter space simulator, and I had an insight of how is inconvenient to land on a body with slow rotation. Usually, when orbiting Earth, I just wait until the planet's rotation brings the landing site into my plane. But when trying to land on the Moon, I need to arrive already almost aligned, because I can't spend days waiting for the rotation.

To make it worse, if I stay a couple of days on the surface, I'll get out of plane from the CSM and will need to correct it on the way back, otherwise I'd have to wait another month.

But that was just my rant... Actually I would like to know how many degrees they had to correct during these alignments. I want to know if I'm running into more trouble than I need.

Thanks.

[ka9q]

The early Apollo landing missions all landed near the equator, so the CSM/LM went into near equatorial orbital planes and there wasn't much worry about the problem you speak of. The later missions landed at moderately greater latitudes (Apollo 15 was the highest at 26 N) so the CSM typically made a plane change the day before the LM ascent stage left the moon. You'll see them listed in the mission events list of a typical mission report.

I haven't really studied perturbations in lunar orbits so I don't know the extent to which they could be used to help change the CSM's orbit to match lunar rotation (and thus minimize the fuel in a plane change). I do know that the perturbations quickly changed the eccentricity (the main reason lunar orbits are so unstable and short lived) and the CSM also had to make allowances for that effect in the longer missions.

[Bob B.]

Jul 21, 2010 12:15:39 GMT -4 Jairo said:

Actually I would like to know how many degrees they had to correct during these alignments.

The size of the plane change is influenced by many factors, such as location of the landing site, inclination of the CSM orbit, and stay time on the surface. However, if all you want to know is the number of degrees of each plane change, that can be determined from the delta-V needed to perform the maneuvers. For example, the following is the lunar orbit phase data for Apollo 12:

history.nasa.gov/SP-4029/Apollo_12g_Lunar_Orbit_Phase.htm

We can see that the CSM plane change took place at an altitude of 62.20 nautical miles, the velocity change was 349.9 ft/sec, and the orbit apolune and perilune were 62.50 n.mi. and 57.60 n.mi. After the burn the perilune changed to 57.61 n.mi., which is insignificant for what we're doing. Since the orbit size was virtually unchanged, we can say that all the delta-V went into changing the plane of the orbit.

The equation for a simple plane change is,

delta-V = 2 * V * sin( theta / 2)

where V is the velocity before and after the burn and theta is the angle change required. We know the delta-V and we can determine the CSM's velocity at the time of the burn from its orbit and altitude. I calculate a velocity of 5,331 ft/sec. We therefore have,

349.9 = 2 * 5331 * sin( theta / 2)

theta = 3.76 degrees

Doing the same thing for the other missions I get:

Apollo 12 - 3.76^o
Apollo 14 - 3.98^o
Apollo 15 - 3.55^o
Apollo 16 - 1.33^o
Apollo 17 - 3.93^o

[ka9q]

Interesting that they were all so close, despite the different inclinations of the CSM orbit (and latitudes of the landing sites). What was different about Apollo 16?

Did the LMs all fly minimum energy ascents, or did they perform doglegs to change inclination or RAAN?

[Bob B.]

Jul 23, 2010 3:50:59 GMT -4 ka9q said:

Interesting that they were all so close, despite the different inclinations of the CSM orbit (and latitudes of the landing sites). What was different about Apollo 16?

I don't know what's up with Apollo 16, though I can certainly imagine scenarios that would result in very little plane change despite the movement of the landing site. Unfortunately, other than orbit altitude, I've be unable to find any information about the CSMs' orbits, such as inclination and longitude of the ascending node. Without additional orbit parameters, I can't figure out the reason for Apollo 16's small plane change.

Jul 23, 2010 3:50:59 GMT -4 ka9q said:

Did the LMs all fly minimum energy ascents, or did they perform doglegs to change inclination or RAAN?

I don't know for sure, but I modeled a simulated ascent of Apollo 17 last year and ended up pretty much on the numbers reported in Apollo by the Numbers. I didn't figure any doglegging, so I assume Apollo 17 didn't either or else I think I would have noticed it in the simulation.

www.braeunig.us/apollo/LM-ascent.htm

It seems far more efficient to let the CSM make the plane change and have the LM perform a minimum energy ascent, rather than land on the Moon the extra propellant needed to make a dogleg. If there was any small out of plane error, that could probably be fixed pretty easily once the LM got to orbit.

[ka9q]

One thing to investigate is the nature of the secular perturbations of lunar orbits. I'm sure you're quite familiar with the usual precessions of earth orbits: the slow change in RAAN (exploited in sun-synchronous orbits) and the rotation of the line of apsides (brought to zero by the 63.5 deg inclination of the Molniya and Tundra orbits). But these are due to the earth's oblateness associated with its fairly fast rotation. The moon rotates 30 times slower, so it can't have a large J2 term like the earth. But it does deviate substantially from a sphere, in part because it's so much smaller. Also because it's smaller, perturbations from the earth and sun are significant.

The Apollo mission planners may have discovered how to make a lunar orbit precess in RAAN so as to partly follow the moon's rotation and lessen the plane change required for a rendezvous with the LM.

At some point in the (distant) future, when we have multiple lunar bases and transportation systems, I'm sure a lot of thought will go into the tradeoffs between landing a mission close to a desired location (or launching it from there) and landing it at a location that's energetically favorable and then traveling across the surface to where you want to be. This will be especially relevant to the bases at the poles that are getting a lot of attention. Suppose you don't care about taking off again; how much more delta-V does it take to do a direct descent to the lunar poles from the earth than to the lunar equator?

Traveling across the lunar surface will be quite an engineering challenge in itself. I never thought the "lunar bus" depicted in "2001" would be a particularly efficient way to do it, as hovering with rockets is pretty wasteful even in a reduced gravity field. But I guess that looked a lot sexier on screen than an ALTV (All Lunar Terrain Vehicle) with big wheels...

[Bob B.]

J2 for the Moon is 0.0002027 while for Earth it is 0.00108263. For a CSM-like low inclination orbit, the RAAN variation due to J2 is about minus one degree per day.

[ka9q]

Interesting, that's actually more than I would have expected. Still, it's not enough to compensate for the moon's rotation, 360 degrees per month or about 13 degrees per day.