Fun with Trigonometry! Or, how to minimize your delta V expenditure when ascending from an airless world

[Norcalplanner]

Genesis of this post:

Back when I was playing my RSS/SMURFF career in 1.1.3, I made a lunar base with refueling infrastructure.  The fully loaded tanker ascending from the surface had a TWR of only 1.57, and my analytical brain wanted to know empirically what angle above the horizon I should point my craft to make it to orbit using the least delta V possible (since the expenditure is so high in RSS).  After playing with some trigonometric functions on the calculator, I was able to figure out that 40 degrees above the horizon was a good orientation for the beginning of the ascent.  And surprisingly, due to maths, I found out that the craft was only "wasting" less than 1/4 of its thrust to stay aloft and avoid unplanned lithobraking.  Because I worked out all the numbers (and just found them again), I thought I would share them here so that others can benefit.

Disclaimer:

I am not a slide rule jockey like some others are on the forum.  However, I do remember some high school math, and know how to use some of the pretty buttons on my calculator.  

The post: 

When it comes to getting to orbit, the mantra goes like this - Vertical velocity is temporary, but horizontal velocity is yours to keep.  

Particularly on airless worlds where atmospheric drag isn't a factor, the corollary becomes - For an efficient ascent, minimize how much delta V you spend on vertical velocity. 

We all know intuitively and/or from experience that thrusting straight up to orbital height, then turning sideways and burning to gain orbital speed results in a very inefficient launch.  It's better to "cut the corner" and ascend at an angle.  There's lots of discussion of gravity turns and ascent profiles from Kerbin, but not much for Mun, Minmus, or other airless worlds.  But exactly how far over should you tip?  That's what this post will let you know.

Here's the big chart with all the data.  I'll do some explanation below.

Degrees above horizon (Alpha)	Horizontal Thrust (Percentage of total)	Vertical Thrust (Percentage of total)	The Trade (Horizontal loss vs. vertical gain)	Efficiency of the Trade	Min TWR to avoid Lithobraking
5	99.6%	8.7%	-0.4% vs. +8.7%	21.8 : 1	11.5
10	98.5%	17.4%	-1.5% vs. +17.4%	11.6 : 1	5.75
15	96.9%	25.9%	-3.1% vs. +25.9%	8.35 : 1	3.86
20	94.0%	34.2%	-6.0% vs. +34.2%	5.7 : 1	2.92
25	90.6%	42.3%	-9.4% vs. +42.3%	4.5 : 1	2.36
30	86.6%	50.0%	-13.4% vs. +50.0%	3.73 : 1	2.00
35	81.9%	57.4%	-18.1% vs. +57.4%	3.17 : 1	1.74
40	76.6%	64.3%	-23.4% vs. +64.3%	2.74 : 1	1.55
45	70.7%	70.7%	-29.3% vs. +70.7%	2.41 : 1	1.41

Degrees above horizon is simply how far above the horizon the craft should point.  This is also the "alpha" angle for the trigonometry.

Horizontal Thrust is how much of the total rocket thrust is being converted into sideways velocity.  Also known as the cosine of alpha.

Vertical Thrust is how much of the total rocket thrust is being converted into vertical velocity (which is temporary and should be gotten as cheaply as possible). Also known as the sine of alpha.

The Trade is just how much of the total horizontal velocity is being forsaken in lieu of an increase in upward velocity. Essentially comparing 1-cos(alpha) to sin(alpha).  

Efficiency of the Trade shows just how cheaply the vertical velocity is being gained because of the way the angles work out.  The number is highest when the angle above the horizon is the smallest, and grows smaller the larger the angle above the horizon becomes.

Min TWR to avoid Lithobraking is what your craft's TWR needs to be (relative to that particular celestial body) to take advantage of that particular ascent angle.  For example, if your craft only has a TWR of 4, then pointing it 10 degrees above the horizon will result in it falling back to the surface and a rapid unplanned disassembly.

Discussion:

So what the chart shows is that, due to the way the math works out, it is most efficient to gain the required vertical velocity by pointing some certain angle above the horizon when ascending.  Yes, the angle will change over time due to changes in mass by burning fuel, and the angle itself will change over time due to the curvature of the body.  But as a general rule, these numbers should hold for the first part of an ascent.  They're also helpful in any situation where atmospheric drag isn't a factor, such as in helping to decide on an angle for an low-thrust orbital insertion stage.  

Speaking for myself, it was amazing just how efficient it can be to "cut the corner" when ascending.  For example, a Munar lander craft taking off with a low TWR of 3 can instantly crank over to 20 degrees above the horizon, terrain permitting, and dedicate fully 94 percent of its total thrust to gaining sideways velocity.  For a TWR of 12, which isn't unheard of on Minmus, the ascending craft can crank it all the way down to 5 degrees above the horizon, expending a mere 0.4% of the total thrust to avoid crashing back into the surface.

TL;DR: Look up the TWR of your Munar lander in the rightmost column, then make an efficient ascent by aiming the craft at the corresponding degrees above the horizon in the leftmost column.    

[Mystique]

Good research.

I'm not really math type (unless it comes to money), but generally tried going as close as horizontal/0 degrees pitch as possible, while avoiding terrain features and keeping vertical velocity somewhat above 0 (totally 0 isn't good idea in low altiltude, since sooner or later something big and rocky will appear in your way). Which your math did totally confirm

[blorgon]

How can the horizontal and vertical thrust percentages add up to more than 100%? Also, I've always been under the impression that launching a projectile at 45° maximizes horizontal distance. Wouldn't it be ideal to launch at 45°, and then go horizontal as soon as your apoapsis reaches the height of your target orbit (assuming we're talking about non-atmospheric bodies)?

[FullMetalMachinist]

5 minutes ago, blorgon said:

How can the horizontal and vertical thrust percentages add up to more than 100%?

Because Pythagoras was a magnificent person. Treat the vertical and horizontal percentages as the two short legs of a square triangle and run it through the Pythagorean theorem. It'll give you something very close to 100%.

[blorgon]

1 minute ago, FullMetalMachinist said:

Because Pythagoras was a magnificent person. Treat the vertical and horizontal percentages as the two short legs of a square triangle and run it through the Pythagorean theorem. It'll give you something very close to 100%.

Except that all of the values in the table add up to well above 100%... You can't have more than 100% of your rocket's total thrust, but the way it's described implies as much:

Quote

Horizontal Thrust is how much of the total rocket thrust is being converted into sideways velocity.

Vertical Thrust is how much of the total rocket thrust is being converted into vertical velocity.

 

[Norcalplanner]

19 minutes ago, blorgon said:

Except that all of the values in the table add up to well above 100%... You can't have more than 100% of your rocket's total thrust, but the way it's described implies as much:

 

A picture will probably help a lot here:

 

Think of the hypotenuse as the thrust vector of the rocket.  The adjacent side is the horizontal component of the vector, and the opposite side is the vertical component of the vector.  If the right angle were to be unfolded, such that the adjacent side (horizontal component) and opposite side (vertical component) were laid end to end, they would indeed have a total length greater than the hypotenuse.  

The "new thinking" of the OP, to the extent that there is any "new thinking" to be had in rocketry or trigonometry, is to calculate opposite/(hypotenuse - adjacent) to produce a number that I'm labeling "Efficiency of the Trade".  It gives you an idea of how much horizontal thrust you're sacrificing at a particular angle in exchange for vertical thrust needed to avoid smashing into the ground.  It's the same concept as combining inclination changes with circularization burns, as you can burn at a diagonal between the two desired vectors to end up where you want to be with a smaller delta V expenditure.  

Edited December 30, 2016 by Norcalplanner
Clarified things