Landing and Takeoff Delta-V vs TWR and specific impulse

[tavert]

So, how did your LaTeX writeup go?

I put a link in at the end of the OP.

Also, an other interesting graph would be payload/fuel mass fraction, for multiple-use ferrys.

Yep. Since that depends on the engine and fuel stats, it's a bit more involved. Check out this tool http://forum.kerbalspaceprogram.com/threads/61659-Wolfram-Web-App-Optimal-Single-stage-Lander-Design-Tool I put together for precisely this purpose.

[Kerano]

Why is the delta v for Eeloo slightly lower than for the Mun, for both takeoff and landing? Shouldn't it be slightly higher, given the larger surface gravity (0.172 g vs 0.166 g) and larger radius (210 km vs 200 km)? All the summary delta v charts I've seen list Eeloo as having a slightly higher delta v requirement than the Mun.

Otherwise great data, very useful.

Edited January 11, 2015 by Kerano

[GoSlash27]

arkie87 and LethalDose did a lot of work along these lines last month.

I was able to apply arkie's results to an analysis of available engines/ tank types and find the T/W ratios at launch that yield the lowest mass and lowest cost for each engine.

Results are here.

Counterintuitively, you will get the lightest and least expensive overall vehicles at surprisingly low t/w ratios, particularly when on larger bodies and when using high Isp engines. Despite costing more in DV (and thus fuel), the mass and cost of fuel/tankage ends up being less than the mass/cost of additional engines required to raise the t/w ratio.

Best,

-Slashy

[KerBlam]

It must be quite difficult to use this method to land accurately, I would imagine, especially if the terrain is mountainous.

[tavert]

Why is the delta v for Eeloo slightly lower than for the Mun, for both takeoff and landing? Shouldn't it be slightly higher, given the larger surface gravity (0.172 g vs 0.166 g) and larger radius (210 km vs 200 km)? All the summary delta v charts I've seen list Eeloo as having a slightly higher delta v requirement than the Mun.

Otherwise great data, very useful.

The difference is down to the sidereal rotation velocity at the surface. You're right that a hypothetical orbit at 0 altitude around the Mun has orbital velocity of 570.7 m/s, whereas around Eeloo it is 595.3 m/s. But Mun's sidereal rotation period is (or was, as of 0.18.2, I don't know whether it's changed since then?) nearly 140,000 seconds, which only contributes 9 m/s to a prograde equatorial launch. Eeloo's rotation period is 19,460 seconds, which contributes 67.8 m/s to a prograde equatorial launch. So if you land from or take off into an equatorial orbit to take full advantage of the surface rotation, overall Eeloo will take less delta-V to land or take off than on the Mun.

It must be quite difficult to use this method to land accurately, I would imagine, especially if the terrain is mountainous.

You can come in at whatever altitude you feel safe, then for the final part of the descent try to maintain a shallow descent while at low speed (still mostly horizontal, but dropping you below whatever safety margin you kept for terrain). You can calculate out the total angular distance that it takes to do this style of landing and predict fairly accurately where you'll land, in terms of longitude difference between when you start your landing burn and the final point of landing.

[Kerano]

The difference is down to the sidereal rotation velocity at the surface. You're right that a hypothetical orbit at 0 altitude around the Mun has orbital velocity of 570.7 m/s, whereas around Eeloo it is 595.3 m/s. But Mun's sidereal rotation period is (or was, as of 0.18.2, I don't know whether it's changed since then?) nearly 140,000 seconds, which only contributes 9 m/s to a prograde equatorial launch. Eeloo's rotation period is 19,460 seconds, which contributes 67.8 m/s to a prograde equatorial launch. So if you land from or take off into an equatorial orbit to take full advantage of the surface rotation, overall Eeloo will take less delta-V to land or take off than on the Mun.

Thanks, that makes sense.

[numerobis]

tavert! How do you do?

I've been toying with the idea of implementing this in a plugin. The impetus is to get an efficient landing, so I can reduce my mass for the single-stage-to-mun-and-back challenge. I'd want to find the highest equatorial flat peak, and stick a landing on it precisely, without wasting much fuel.

[tavert]

Hiya numerobis, been a while.

Shouldn't be too hard, you probably don't need the fancy variable-step integrators that Mathematica uses here. Simple RK might work well enough with small steps. Interesting thing to play with here is the variable transformation of whether you want to integrate velocity as a function of time, or the other way around.

[Thunderous Echo]

Question/Challenge/Request:

Is there an equation that, given a certain initial thrust, specific impulse, and mass of a parent body:

Starts at a certain horizontal velocity, at a certain altitude (equatorial radius of the parent body), with a vertical velocity of zero, and

Ends at a certain horizontal velocity, at a certain altitude, with a vertical velocity of zero, and

can calculate how much delta-v (or fuel mass) this change of horizontal velocity while maintaining zero vertical velocity?

Because, if there is, then one could calculate how much delta-v (or fuel mass) that a multi-stage landing or takeoff would take by plugging in such a formula for each stage.

(There may also need to be a formula for how much horizontal velocity you can change with a certain amount of delta-v).

[tavert]

Not exactly a closed-form equation, but if the initial and final altitudes are the same and you want to stay at exactly zero vertical velocity through the entire burn, you can use the same differential equation just with custom initial and final horizontal velocities instead of the specific values I used here (which were based on orbital / rotation speeds).

[Thunderous Echo]

Oh wait.

OOOOOOOOOOOOOOHHHHHHHHHHHHHHHHHHHHHHHHH.

Oops.

[Kobymaru]

Hello friends of Landing math, hello @tavert  Unfortunately, your awesome derivation document and landing/takeoff calculation code have disappeared from your dropbox  Would it be possible for you or for someone to upload a copy somewhere? I think this kind of math is very useful. It would be much appreciated.

[Streetwind]

That probably happened because Dropbox decided to remove a feature called "public folder". Previously, you had this special folder where you could just dump a file into and publicly link to it from the internet. A few days ago, this was converted into a normal folder, and all files in it are no longer accessible. Instead, a user must manually create a sharing permission for each individual file, and the resulting share cannot be hotlinked to, but rather always redirects to a "sign up for Dropbox now" begging page first unless you happen to already have an account and are logged in.

In my case, about 7 years of writing game guides and tutorials and participating in forum discussions with screenshots, spreadsheets, patches and configs across more than five different games just lost all of its linked content. Fancy, ain't it?

Well, at least tavert's file hasn't been lost. It's just no longer publicly accessible right now. He will be able to link it again. ...Or he would, if he hadn't gone inactive in 2015. =/