2.2 Launch Compute

Orientation

We start our mission on a lunar base called Brighton Beach. The ship class is called Shuttle-A, and its callsign in this scenario is SH-03. Our target is an orbiting space station flying high overhead, identified in our navigation systems as ESS. 

Although the ship is carrying a 120 tonne payload, it will not take too much effort to get it off the pad, as the gravity on the moon is 1/6th of what you will find on Earth, and this ship class is capable of performing interplanetary flights. 

For contextual information about our ship, our target, and the body we will orbit, we can pull up information tables from the Orbiter menus:

In the cockpit view, we can access plenty of information through the left and right display windows, called Multi Function Displays (MFDs). 

On the left MFD we are currently showing Orbit, and on the right MFD we are showing the Surface Map. In both windows we have entered the space station "ESS" to be our target, and it shows up in yellow color. In the map MFD we can see the surface coordinates of the lunar base at 41.12°N, 33.45°W, and we can see the orbital track of the space station in yellow. In the Orbit MFD we can see information of the target's orbit and current position printed in yellow. 

Note carefully: in orbit MFD, we have set the reference mode is set to EQU, meaning the frame of reference is the moon's equator, i.e. normal to the moon's axis of rotation. (The moon does rotate like the Earth, but much slower). We can see that the space station is orbiting roughly eastbound, at an inclination of 26.87° relative to the moon's equator. Also take note of these key values for the space station:

• eccentricity =  0.0001 (almost perfectly circular)

• longitude of ascending node (LAN):  154.20°

• apoapsis (ApR):  2,234 km from the center of the moon 

• periapsis (PeR): 2,233 km from the center of the moon  

• orbital period (T): 9,472 seconds

Note very carefully: the longitude (E/W) values seen on the map correspond to positions on the surface of the moon; therefore, they do not necessarily correspond to the longitude of ascending node. The longitude of ascending node is measured from a vector that does not rotate with the moon; its direction is defined by the cross product of the moon's axis of rotation and the vector normal to the ecliptic plane. So if you look closely at the space station's track on the map, you will notice that at the point where it crosses the moon's equator, the longitude does not match the LAN value. 

Launch Plan

Our plan is to launch on a launch heading of 090°: due east relative to the moon's equator. Using this launch heading means the inclination of our initial orbit will match the latitude of the moon base we are launching from, which is 41.12°. 

Note very very carefully: although we know the inclination of our target orbit, and also know the expected inclination of our launch orbit, we do not know the relative inclination of these two orbital planes. The relative inclination is not simply the difference between their two inclinations. Consider two orbital planes with identical inclinations: their relative inclination will only be 0° if their ascending nodes have matching longitudes. Computing these values is quite a lot of work, requiring techniques beyond what I outlined in the theory roadmap; but luckily in Orbiter we have an MFD called Align Planes which does it for us. And even sitting on the launch pad, it is already computing a value as if the ship orbit was at an orbit inclination of the base's latitude. This is why I am using the 090° launch angle: it lets me use this RINC value to estimate the burn required for orbital plane alignment. 

We will launch our flight when the space station is directly south of us. The plan is to perform a rapid acceleration by firing the main engines for about 2-3 minutes, and then let the ship coast along a curved trajectory until the apex ("apoapsis"). We will plan our apex to have the same altitude as the target space station. When the ship arrives at the apex, it will not have enough speed to maintain this orbit, so it will immediately start descending back down towards the lunar surface. So, when the ship reaches the apex, we will fire up the engines a second time to boost the speed enough to allow it to maintain a circular orbit. 

Launch Procedure

The launch procedure will be as follows:

• use the hover engines to get some vertical clearance from the lunar base and surrounding terrain; raise the landing gear

• rotate the nose to point to heading 090°

• cut the hover engines

• pitch the nose up to 10° above the horizon

• fire the engines at 80% of full power for 170 seconds (just under 3 minutes). Keep the nose of the ship at 10 degrees above the horizon until the engines are cut off. 

• coast until apex ("apoapsis")

Launch Trajectory Computation

To compute the launch trajectory I am running a little Python script. The three input variables are:

• pitch angle

• engine power

• burnout time

I experimented with a few combinations until I arrived at 10° pitch, 80% power and 170 seconds. Note that there are other valid combinations which could produce the same end result. The code is shown below.

Imports

Estimate fuel required for takeoff hover

Here we make a quick estimate for the amount of fuel required to use the hover engines to clear the launch pad. This is just a simple estimate where we assume one minute of hover thrust, at a constant thrust setting where the thrust equals the weight of the spaceship:

This gives:

Loop to simulate 2D trajectory in orbital plane

Throw results into a dataframe

We throw the results into a dataframe:

Print and Visualize Results

Here are the summarized computation results:

Below we plot the trajectory. I would point out the following:

• the target orbit, i.e. the space station's orbit, is shown in red

• note how shallow the orbit insertion angle is

• the engines were fired for under 3 minutes, but the total time from launch to apex is almost one hour. On the moon we are lucky that there is no atmosphere to push through