Tutorial Reaching the Moon with basic math

[sorindafabico]

This tutorial is intended for begginers.

Some days ago I was wondering a way to go from the Cape to the Moon using less math as possible, avoiding IMFD, TransX and LTMFD, and avoiding the plane alignment burn shown in the stock Orbiter tutorial.

You can't simply launch a vehicle to a parking orbit perfectly aligned with the Moon from latitudes greater than 28 degrees, so we need to make an off plane transfer.

The solution I found is using OrbitMFD and two special windows that occur twice a month, when the Moon crosses the ecliptic plane. The idea applies to both vessels on the ground and already in LEO, if the crew has enough oxygen to wait for the ship's nodes align with the Moon's nodes.

This method is enough to make you reach the sphere of influence (SOI) of the Moon.

Follow it step by step. If you found it confusing, ask me to clarify.

The MFDs we'll use are:
- Orbit MFD (mandatory)
- Launch MFD (recommended)
- BurnTime MFD (recommended)

You'll need basic knowledge in these MFDs (anyway, I suggest you to learn them in detail since they are very useful).

Step 1 - on the ground:

First, open Orbit MFD and press DST if it is showing PeA instead of PeR. You need PeR. The frame of reference needs to be ecliptic - press FRM until "Frm" be ECL (top right of the MFD screen). Hit TGT and choose the Moon.

The first number you need to get is a rough estimative of the time of flight (ToF). The ToF is given by:

Here, a1 and a2 are the semi-major axis (SMa in Orbit MFD) of the Moon and your intended parking orbit, G is the gravitational constant and M is the mass of the Earth.

After you got the ToF (~425000 seconds, get it exactly since the Moon's SMa changes with time due to perturbations), you need the mean motion of the moon. The mean motion is simply 2*pi/T (radians) or 360/T (degrees). T is the orbital period of the Moon showed in Orbit MFD in seconds.

Then, with (ToF)*(mean motion) = DL you have the angle the Moon would be travelling if its orbit was circular while you will be coasting (after the TLI). You'll get something near 65.7 degrees.

We want to reach the Moon when it crosses the plane of the ecliptic, i.e., when its Mean Longitude (MnL in Orbit MFD) is equal to the longitude of one of the nodes. Orbit MFD shows the Longitude of Ascending Node (LAN). The other node is LAN ± 180 degrees. For the sake of simplicity, we'll call both nodes "LN". Use the next avaliable node when you're playing, the AN has nothing special here.

We have the longitude where we'll encounter the Moon (LN) and the longitude the Moon will be when we'll do the trans-lunar injection (LN - DL). Now, we need to define the launch parameters.

First, get a rough estimative of the time you'll spend between the launch and the TLI, let's say, some 7000 seconds. Use the mean motion of the Moon to calculate the mean angle the Moon will travel in these 7000 seconds and subtract it from (LN-DL). You now have the longitude where the Moon needs to be (roughly) when you launch.

So, you have the launch time. Then, you need to launch for a parking orbit with the same ecliptic LAN (or ± 180 deg) of the Moon. You are free to choose the inclination and the SMa - I suggest a circular orbit at 220 km and the inclination that minimizes the difference between the optimal time for launch and the time when the Moon passes at the longitude where it needs to be when you launch (if you don't have fuel constraints). The best way to do this is with Launch MFD.

If you know how to use Launch MFD or if you want to use another method to reach LEO, jump to step 2.

Select Launch MFD. Hit ALT and choose the altitude of your parking orbit. Hit TGT and enter inclination and LAN values as "INC LAN c". You need to write the letter c in order to tell the MFD you want the ecliptic plane. Then, launch your vessel and click AP to give the control to the autopilot. DEF allows you to toogle the pitch AP on and off.

Step 2 - LEO and TLI

Once in LEO, you need to estimate the delta-v. First, let's find the radial distance of the Moon at the encounter. The distance r is:

A is the SMa of the Moon, e is the eccentricity (Ecc) and theta is the [ame="http://en.wikipedia.org/wiki/Orbital_elements"]True Anomaly[/ame] (TrA), the angle between its position and the position of its periapsis. The angle theta you need to put in the formula is the angle between the longitude of the node you're going to reach the moon and the longitude of the periapsis (LPe) of the Moon (LN - LPe).

With r we can do a better estimative of the delta-v than with a. This is critical, since the difference can send you far enough from Moon's SOI if the Moon's apoapsis and periapsis are close to the nodes.

To calculate the delta-v, use:

Here, rm is the value you just calculated, rs is the radial distance (Rad) of your ship.

Now we have the delta-v and the MnL we need to be when firing the engines (LN - DL ± 180 deg). Time to open Burn Time MFD.

First, hit MD until "Time to manual start" appears. Hit dV and enter the value of delta-v you found. BTMFD will give you a critical information: "Half deltaV time". You need to start the TLI before (LN - DL ± 180 deg), because we can't apply delta-v instantly. Calculate YOUR mean motion (360/T_ship), multiply it by "Half deltaV time" and subtract the result from (LN - DL ± 180 deg). Then you have the real MnL where you'll start the burn. Look for it in Orbit MFD and hit BRN when it's time. Don't forget to burn prograde.

Then, it's time to rest and prepare to the LOI. There, you can use the stock Orbiter tutorial about going to the Moon to learn how to do a lunar orbital insertion.

I made this twice, in both times I reached the Moon's SOI.

 

[mjf]

great idea. I'm now much more friendly to add-ons, but I remember when I first started using Orbiter, I didn't want to bother with installing any fancy MFDs.

This approach is a really good one for people who want to try the "bare-bones" approach; if you think about it, one could then gradually start using the more advanced MFD add-ons but still run the same process, in order to actually learn what the MFDs are doing...

 

[richfororbit]

Interesting, when I looked into the technical side of various keywords to do with orbiting, I just completely switched off.:blink:

For example, if I wanted to put the Mars probes into orbit like Maven or MRO, I'd have to do that, and then tweak the orbits to essentially create not even a percentage of what it would be like to do a burn to capture into Martian orbit.:uhh:

Travelling to the Moon or Mars is a long way off.

 

[sorindafabico]

Interesting, when I looked into the technical side of various keywords to do with orbiting, I just completely switched off.:blink:

For example, if I wanted to put the Mars probes into orbit like Maven or MRO, I'd have to do that, and then tweak the orbits to essentially create not even a percentage of what it would be like to do a burn to capture into Martian orbit.:uhh:

Travelling to the Moon or Mars is a long way off.

The principle behind the idea is [ame="http://en.wikipedia.org/wiki/Orbital_node"]node[/ame] alignment. If your nodes and target's nodes are aligned, you only need to time the transfer burn so that you and the target will arrive in the same node at the same time. Zero relative inclination isn't mandatory, but the relative inclination is proportional to the spent fuel when doing orbital insertion (unless it matches with your desired orbital configuration around the Moon, of course).

Also, if the relative inclination isn't too big and the nodes are well placed, you can try a mid-course plane alignment. But this is not the same as reach the Moon's SOI with only one burn. :tiphat:

For planetary transfers the math is more compicated.

 

[richfororbit]

Well going to Mars in an MPCV is out of the question, perhaps the space probe, paper designed by the British planetary institute could go there. But that is another subject as you have stated.

Ofcourse I'd be going from low Earth orbit in an MPCV to the moon.

---------- Post added 04-17-15 at 08:36 PM ---------- Previous post was 04-16-15 at 10:21 PM ----------

On the subject of when the Moon is closest to our planet, what about the years that those Astronauts went there, doesn't the simulator work to that precision, I imagine so.

The moon appears to be closest during March 2001 than July 1969.

 

[sorindafabico]

On the subject of when the Moon is closest to our planet, what about the years that those Astronauts went there, doesn't the simulator work to that precision, I imagine so.

The moon appears to be closest during March 2001 than July 1969.

The [ame="http://en.wikipedia.org/wiki/Apollo_Guidance_Computer"]Apollo computer[/ame] used numerical integration methods similar to IMFD's Map program and it had better precision than OrbitMFD, because OrbitMFD performs a 2-body calculation and doesn't accounts for perturbations caused by other bodies.

The 2-body calculation performed by OrbitMFD is enough for the procedure I described above, but if you want more precision - things like a well defined orbit around the Moon without course corrections, move to LTMFD.


About the distance of the Moon: it's not so critical to be at perigee or apogee. You can verify this in the last equation of the OP (the delta-v one).