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When Apollo astronauts departed the lunar surface in the Lunar Module's Ascent Stage, it appeared to rise vertically ($\beta$ = 0) for ~10 seconds and then begin the "tipover" (videos of Apollo 15, 16 and 17), the program of increasing $\beta$ ( = $\pi/2 - \gamma$) with time, transitioning from vertical flight to orbit, where $\beta$ was just short of $\pi/2$ at ASP shutdown.

In the videos, especially the out-the-window video, it is apparent that the $\beta$ vs. time program is not a smooth pitch-over but instead has discrete (short) intervals of increasing $\beta$ with intervals of nearly-constant $\beta$ between.

QUESTION: Given the physical and performance parameters of a fully loaded Ascent Stage, what would be the optimal time profile of $\beta$ to minimize propellant use attaining the initial (@RussellBorogove's answer to another question specifies 18 x 87 km) lunar orbit?

Assume launching from a spherical moon and from central Mare Tranquillitatis, so no mountains around, and no margins.

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    $\begingroup$ Optimal for the real shape of the moon and some minimum safe miss-the-mountains distance, or for a perfectly spherical moon and no error margins accounted for? $\endgroup$ – Russell Borogove Jul 22 '18 at 19:10
  • $\begingroup$ @uhoh They are already linked. You just have to click on the flight number of the Apollo mission (15, 16, 17). I tested the links just now and they worked fine. (I posted this same comment about 10 minutes ago but it seems to have disappeared; try, try again!) $\endgroup$ – Tom Spilker Jul 23 '18 at 1:00
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    $\begingroup$ I've been putting off investing in a pair of glasses for about a decade. It's becoming an increasingly difficult position to maintain ;-) $\endgroup$ – uhoh Jul 23 '18 at 1:06
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Given some simplifying assumptions (constant thrust, constant gravitational field, flying in vacuum, over flat horizontal surface), which aren't too badly undermined by the lunar ascent case, the linear tangent steering law is known to give optimal orbital insertion:

$$\tan \theta = A \cdot t + B$$

i.e. the tangent of the thrust pitch angle changes linearly from the point at which insertion guidance starts until circular orbit is achieved. The trouble is that A and B aren't easily derived!

Using a home-brewed simulation using the initial state figures from the Apollo 11 mission report and a very crude, ad-hoc sampling of possible values for A and B, the best-case ∆v-remaining-at-insertion figure I have so far found was for A = -0.002100, B = 0.8040, that is to say:

$$ \theta = \arctan ( { 0.8040 - 0.0021 \cdot t } ) $$

Where $t$ is the time in seconds from when the guidance program takes over. My sim executes a fixed vertical ascent from liftoff until vertical speed reaches 12 m/s, like the actual ascent stage, before switching to this guidance algorithm.

Note that this is the desired thrust angle, thus the commanded vehicle pitch angle rather than the flight path angle $\gamma$ that you asked for, though the two should tend to converge towards the end of the run. If you care about the distinction there's probably some calculus available to derive $\gamma$ from $\theta$, or I could pull the data out of my sim.

The simulator state at insertion and cutoff was a 18.14km x 87.58km orbit, altitude 18.36km, lunacentric velocity 1687.57 m/s, remaining ∆v available 182.18 m/s. Pitch program takes over at T+7.21 seconds and cutoff is at T+436.66 seconds (about 1.5 sec earlier than nominal and about 1.5 sec later than Apollo 11's actual performance). 2218 kg (4889 lbs) of propellant was expended; this compares favorably with the mission planning report, with a nominal ascent requiring 4966 lbs, but it's unclear if the difference is due to unrealistic aspects of my simulation or an actual improvement in the trajectory. I was unable to find a pitch schedule for the actually-flown ascent; I could make a fair comparison in the sim if I had that.

According to this very thorough guide to the LM descent and ascent programs, the ascent guidance updates on a two-second cycle, which causes the discretization of the pitch observed in the videos.


: For instance: my simulated vehicle snaps instantly to a commanded attitude, whereas the real LM does not. The real LM ascent engine's nozzle and throat erodes during flight, changing its specific impulse slightly, whereas my sim uses a fixed intermediate value for specific impulse for the whole ascent.

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    $\begingroup$ Hmmm, in my case it is actually attitude, not path angle— I’m used to the atmospheric Earth orbit ascent case where the two are closely matched. I think the tangent law applies to thrust direction, rather than flight path; let me review. Good catch. $\endgroup$ – Russell Borogove Jul 31 '18 at 3:28
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    $\begingroup$ My $\gamma$s are now $\theta$s and things should be somewhat less wrong now. $\endgroup$ – Russell Borogove Jul 31 '18 at 3:38
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    $\begingroup$ Initially I was trying to do this analytically with 1/r^2 gravity and a spherical moon (and with some of the same simplifying assumptions, such as constant thrust & specific impulse, instant attitude changes) but I got an intractable (to me!) differential equation. So I started over with a home-brew orbit integrator. I wasn't familiar with the tangent law (thank you for that steer!) but what I wound up with looks fairly similar to an arctan. I also get burn times (and thus propellant use) a bit less than the Apollo 11 profile. $\endgroup$ – Tom Spilker Jul 31 '18 at 4:17
  • $\begingroup$ I want to "play simulator" too! What initial mass, thrust, (and perhaps ISP) are you guys using, roughly? $\endgroup$ – uhoh Jul 31 '18 at 4:41
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    $\begingroup$ @uhoh I'm using Mo = 4672 kg, Isp (avg) = 311 s, F = 15569 N (3500 lbf) for an M-dot of 5.1048 kg/s, GM-moon = 4902.8 km^3/s^2, R-moon = 1737.1 km. $\endgroup$ – Tom Spilker Jul 31 '18 at 4:53

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