Maths (very very frightening)

[GoSlash27]

3 hours ago, Gojira1000 said:

Oo, OK that's just opened up a can of potential worms in my planning of orbital stations.

Gojira1000,

 I actually have some recommendations for this. I'll start a new thread about it so as not to drag this one further off.

Best,
-Slashy

[Yasmy]

I'm a bit late to the party, but here's a chart I made a while ago just for these kinds of questions.

My answers differ slightly from Slashy's.  421 m/s direct vs. 464 via dropping down to low Duna.

Interplanetary delta-v cost for direct from moon vs. dropping down to parent

P.S.: This table is for starting or dropdown passes 30 km above airless moons, 50 km above airless planets, or a multiple of 10 km above bodies with an atmosphere

Edited April 9, 2016 by Yasmy
Responding to Plusck below

[Plusck]

2 hours ago, Yasmy said:

I'm a bit late to the party, but here's a chart I made a while ago just for these kinds of questions.

My answers differ slightly from Slashy's.  421 m/s direct vs. 464 via dropping down to low Duna.

Interplanetary delta-v cost for direct from moon vs. dropping down to parent

Excellent. Bookmarked.

Do you remember what orbits you started from for these calculations?

[GoSlash27]

Yasmy,

 This is a really cool chart!

 I'm curious as to why your calculations consistently work out to less DV than mine, even direct from low parent orbit.

Best,
-Slashy

[OhioBob]

22 hours ago, GoSlash27 said:

Hey y'all! I just came up with a simple formula to determine the optimal radius to begin the transfer burn for minimal DV:

r=2u/Vxs² 

I ran this on a few examples in my spreadsheet to make sure it checks out. In the above example, the minimal DV altitude for Duna to Kerbin direct would be 563,291m.

For Kerbin to Duna, it would be 7,775,080m.

Best,
-Slashy

That reminds me of this conversation from 6 months ago,

http://forum.kerbalspaceprogram.com/index.php?/topic/122445-oberth-effect/

The equation you derived is the same one we came up with back then.  It's called the gate orbit.

[GoSlash27]

OhioBob,

 It's always comforting to have this sort of confirmation. If I show up at the party wearing the same dress as you, I must be doing it right

Best,
-Slashy

[Yasmy]

6 hours ago, GoSlash27 said:

I'm curious as to why your calculations consistently work out to less DV than mine, even direct from low parent orbit.

I don't know, Slashy.  I checked your equations, and they look approximately correct to me. They are approximate though, correct in the limit that R -> infinity, where R is the radius of the SOI. That could account for some errors.

In particular, you say that V_esc = sqrt(2) V_orb. (Cute.  I never noticed this before.) That's not exactly true in the SOI approximation. Using the vis-viva equation at the SOI boundary of radius R gives the semi-major axis for escape at zero velocity:

0 = v² = mu (2/R - 1/a)   ->  mu (2/R) = 1/a

Now at some radius r < R, on the same orbit, v_esc² = mu (2/r - 1/a) = mu (2/r - 2/R) = 2 (V_orb²(r)-V_orb²(R)).

But of course this is often a tiny correction, particularly for planets. For moons, it might be a more significant error. In fact, for some light moons, you can often exit on a fairly low eccentricity elliptical orbit instead of hyperbolic orbit because the SOI is too small.

Having said all that, I still can't say why we get such different results.  Perhaps my orbital data is bad or my code has a bug...

[GoSlash27]

Yasmy,

 Now that you mention it, I think that's exactly what it is; Vesc= Vorb*sqrt(2) is true for n- body, but not correct for KSP because it cuts off gravity at the SoI boundary. That would account for the difference.

 

[OhioBob]

Yeah, in the real universe we use hyperbolic excess velocity, which is the velocity remaining after escaping a planet's gravity.  Technically, it is the velocity remaining after reaching infinity, but for all practical purposes, once we've moved a great distance away we have escaped.

V_∞² = V_bo² - V_esc²

where V_∞ = hyperbolic excess velocity, V_bo = burnout velocity, and V_esc = escape velocity.

In KSP's patched conics, the equivalent to V_∞ is the velocity remaining after crossing the sphere-of-influence.

V_soi² = 2μ * (1/R_soi - 1/R_bo) + V_bo²

where V_soi = velocity at the SOI, V_bo = burnout velocity, R_soi = radius of the SOI, R_bo = radius at burnout, and μ = gravitational parameter.

Note that V_soi = V_∞ when R_soi = ∞.

I usually do my calculations using V_∞ and V_esc because the math is easier, realizing that it produces a value a little higher than needed for KSP.  It's a little conservative, which is usually not a bad thing.
 

Edited April 10, 2016 by OhioBob

[Streetwind]

I'm digging this back up because I want to understand this better, and to learn how to actually solve it in practice.

My first question is in regard to the gate orbit equation, r = 2µ / V_∞². Where do I actually get that V_∞² parameter from? Looking it up, Wikipedia helpfully reminds me that V_∞² = C3, and that C3 = µ / a, where a is the semimajor axis of the hyperbola. Which means that you could simplify the gate orbit equation to r = 2µ / (µ / a) = µ * a. Which is all fine and dandy, except that:
- I have no clue how to calculate the SMA of a hyperbola.
- Even if I had that, I still wouldn't know how to solve this equation in practice.
Like, isn't the shape of the resulting hyperbolic orbit, and thus its SMA, totally dependant on your starting orbit? If so, how can the equation spit out the ideal starting orbit? That doesn't make sense. The only possible explanation therefore is that I got this all completely wrong.

...Halp?

 

Edited April 22, 2016 by Streetwind

[Plusck]

2 hours ago, Streetwind said:

I'm digging this back up because I want to understand this better, and to learn how to actually solve it in practice.

My first question is in regard to the gate orbit equation, r = 2µ / V_∞². Where do I actually get that V_∞² parameter from? Looking it up, Wikipedia helpfully reminds me that V_∞² = C3, and that C3 = µ / a, where a is the semimajor axis of the hyperbola. Which means that you could simplify the gate orbit equation to r = 2µ / (µ / a) = µ * a. Which is all fine and dandy, except that:
- I have no clue how to calculate the SMA of a hyperbola.
- Even if I had that, I still wouldn't know how to solve this equation in practice.
Like, isn't the shape of the resulting hyperbolic orbit, and thus its SMA, totally dependant on your starting orbit? If so, how can the equation spit out the ideal starting orbit? That doesn't make sense. The only possible explanation therefore is that I got this all completely wrong.

...Halp?

 

Maybe you're confused about what V_∞ is?

V_∞ is the excess velocity you need, in addition to (or subtracted from) Kerbin's own orbital velocity in order to do a Hohmann transfer to another planet. To get that, you use your altitude above the sun to get orbital velocity at current orbit, then obtain your required Pe velocity where the SMA corresponds to the elliptical transfer orbit between original and target orbit.

From wikipedia: C3 is the characteristic energy, = −GM/a, where a is the semi-major axis, which is infinite for parabolic orbits.

Note that there is a minus sign there This is important because C3 is only positive if you are not confined in an orbit. C3=0 is a perfect parabola, i.e. escape with 0m/s at infinity, therefore -GM/∞ =0. Wiki should add that a is negative for hyperbolic oribits.

However this is not the same a as the gate orbit's a. The a in "−GM/a" is the SMA of the resulting hyperbolic orbit. The "a" in the other equation is the a of the departure gate orbit.

Edited April 22, 2016 by Plusck

[Streetwind]

Very very confused, yes =/ Abstract maths was never a strong point of mine.

 

39 minutes ago, Plusck said:

From wikipedia: C3 is the characteristic energy, = −GM/a, where a is the semi-major axis, which is infinite for parabolic orbits.

Yeah, that's precisely what I wrote (apart for using µ for GM). But isn't this a hyperbolic case, not a parabolic one?

As for the minus sign, Wikipedia doesn't have it. It specifically states C3 = GM / a, not -GM / a. How come? I mean, Wikipedia can be wrong, but the chance is usually very low. Especially for things like this which are easily verified facts.

 

39 minutes ago, Plusck said:

The a in "−GM/a" is the SMA of the resulting hyperbolic orbit. The "a" in the other equation is the a of the departure gate orbit.

...So that means I cannot use r = µ * a to solve this equation, because both r and a are unknowns? Is that why everyone uses r = 2µ / V_∞² instead?

 

39 minutes ago, Plusck said:

To get that, you use your altitude above the sun to get orbital velocity at current orbit, then obtain your required Pe velocity where the SMA corresponds to the elliptical transfer orbit between original and target orbit.

...I think I know some of those words. *stares blankly*

 

Edited April 22, 2016 by Streetwind

[Plusck]

2 hours ago, Streetwind said:

Very very confused, yes =/ Abstract maths was never a strong point of mine.

 

Yeah, that's precisely what I wrote (apart for using µ for GM). But isn't this a hyperbolic case, not a parabolic one?

As for the minus sign, Wikipedia doesn't have it. It specifically states C3 = GM / a, not -GM / a. How come? I mean, Wikipedia can be wrong, but the chance is usually very low. Especially for things like this which are easily verified facts.

 

...So that means I cannot use r = µ * a to solve this equation, because both r and a are unknowns? Is that why everyone uses r = 2µ / V_∞² instead?

 

...I think I know some of those words. *stares blankly*

 

https://en.wikipedia.org/wiki/Escape_velocity

The minus sign is definitely there on the escape velocity page.

The C₃ equation works for all orbits: if the orbit has a small semi-major axis, then that "characteristic energy" C₃ will be a negative number, tending towards -µ (or -GM) as the radius tends towards 1. If it has a large SMA, it will be a small negative number, tending towards zero as the orbit inceases in size. At that "perfect escape" moment, when the orbit is parabolic, SMA will be infinite and therefore C₃ will be zero. As the orbit becomes hyperbolic, then there is some residual energy at infinity, and that is what you're using to go to another planet after escaping the planet of departure.

"...So that means I cannot use r = µ * a to solve this equation, because both r and a are unknowns? Is that why everyone uses r = 2µ / V_∞² instead?"

Yes - sorry I actually introduced an added confusion in there, I should have said that r and a are in two different orbits.

However, you can't use r = µ * a because you made a mistake in the equations. 

r = 2µ / (-µ / a)

r * -µ / a = 2µ

-rµ = 2µa

r = -2a

So yes, sure, you can determine the semi-major axis of the optimal hyperbolic trajectory - or alternatively the optimal C3, that you can get when starting in a circular orbit with radius r. Or the optimal gate orbit r for a given hyperbolic a.

And the square root of that C3 is your suplus velocity on leaving the SOI.

 

..."stares blankly"...

If you go from Kerbin to Duna, say, you need to burn about 1050 m/s from LKO. Your orbital velocity at 75km then becomes (approximately) 2300+1050 = 3350 m/s. If you only burned 920 m/s, you'd end up with an orbital velocity at 75km of 3220 m/s and an Ap at the SOI edge with essentially zero velocity. However, because you're going 100 m/s faster, you escape with quite a lot of excess velocity.

You can get the SMA for that non-escape burn from v² = µ * (2/r - 1/a). Since we know v, µ and r (r=75km+Kerbin radius of 600km), we can turn that around to get: a = µ/(2µ/r-v2), giving 36,941km (which, given that r at Pe is 675km, puts Ap at almost 2xSMA, or around 72Mm which is quite a bit beyond Minmus's orbit).

If we increase v at 75km up to 3350 m/s (Duna escape burn), we get an "a" of -4656032 and a C₃ of 7584500. Therefore V_∞² = 7584500 and V_∞ = 870 m/s.

So when we leave Kerbin's SOI, we have a residual velocity of 870 m/s, which then get's added to Kerbin's orbital velocity (9284 m/s) at Kerbin's solar altitude of 13,600Mm.

Going back to that same equation — v² = µ * (2/r - 1/a) — we can get the SMA again from knowing v, µ (were µ = GM_Kerbol = 1.1723x10¹⁸) and r = 13600Mm, giving a = 16,920Mm.

The SMA of a transfer orbit from Kerbin is halfway between Pe at Kerbin and Ap at the destination orbit, so our solar Ap with that burn would be 20,240Mm... which is higher than Duna's Pe and lower than it's Ap, and almost exactly at its SMA (20,726Mm).

So, with an excess velocity of 870 m/s from our escape burn of 1050 m/s, we can indeed get to Duna.

 

Edited April 22, 2016 by Plusck

[Streetwind]

@Plusck, I really appreciate your effort in explaining this to me, but it's probably not something that wants to go into my head... I'm probably missing some more fundamental stuff that would be necessary to understand these calculations. Like, I spent the past two days scratching my head in utter confusion, before it finally dawned to me that the equation you base the second half of your post on is actually something that has a name too, and isn't just a string of numbers and variables that happened to be relevant here. I was completely unfamiliar with the vis viva equation until I googled it just now. I'm fairly sure that there are more such things that I'm missing, which is why I fail to "get it".

If I could trouble you one more time - would it be possible to put these calculations into a spreadsheet? One that a layman like me could use to calculate gate orbits in an arbitrary solar system, just by putting in things like the relevant GM's and SMA's which I can look up ingame? Or is that problem too complex?

[Plusck]

4 minutes ago, Streetwind said:

@Plusck, I really appreciate your effort in explaining this to me, but it's probably not something that wants to go into my head... I'm probably missing some more fundamental stuff that would be necessary to understand these calculations. Like, I spent the past two days scratching my head in utter confusion, before it finally dawned to me that the equation you base the second half of your post on is actually something that has a name too, and isn't just a string of numbers and variables that happened to be relevant here. I was completely unfamiliar with the vis viva equation until I googled it just now. I'm fairly sure that there are more such things that I'm missing, which is why I fail to "get it".

If I could trouble you one more time - would it be possible to put these calculations into a spreadsheet? One that a layman like me could use to calculate gate orbits in an arbitrary solar system, just by putting in things like the relevant GM's and SMA's which I can look up ingame? Or is that problem too complex?

Actually, the simplest thing would just be to give you the spreadsheet I used for these calculations.

It's unfinished, and ugly, and has a whole bunch of stuff that has nothing to do with this question but which I was trying (and failing) to work into a graph.

It's also from a French version of Excel, and I have no idea how well Excel translates its spreadsheet functions. It would be totally stupid if it didn't, but one never knows...

But anyway, here it is.

[Streetwind]

It actually translated perfectly fine

Now I just need to figure out how to use it. So the green area is where I need to change stuff, and outside of it I shouldn't change stuff, right?

What would I need to do to change the source or the destination body, respectively? And P5 has the gate orbit altitude, right? ...Is that above planet surface, or above the planet center?

[Plusck]

45 minutes ago, Streetwind said:

It actually translated perfectly fine

Now I just need to figure out how to use it. So the green area is where I need to change stuff, and outside of it I shouldn't change stuff, right?

What would I need to do to change the source or the destination body, respectively? And P5 has the gate orbit altitude, right? ...Is that above planet surface, or above the planet center?

I named a few cells like "variable u" (meaning µ) and "variable a" and suchlike for the graph function - so the named variables all correspond to orbit around Kerbin. However, since the second part was dealing with 870m/s suplus velocity heading to Duna, I needed to stick the sun's info in there (like its mass in I2, giving GM_kerbol in H2). J2 is Kerbin's orbital radius, which is also its SMA.

For the graph, the idea was to enter data in B3 B6 and B7 to get different graphs.

J4 is actually just the same as "variable r2" (i.e. altitude above Kerbin + Kerbin's radius), but since I wasn't wanting to mess with the graph bit I just added it in again for a 75km orbit.

K4 is velocity at Pe. If you use B6 and B7, you'll get the orbital velocity at Pe and Ap for that elliptic orbit. However, N6 and N7 also give you the orbital velocity for a circular altitude at that orbit. So you could change J4 to read "=variabler2" and change K4 to read "=N7+[whatever escape burn you want]" and it would be more useful.

L4 gives the SMA that would result from that velocity at Pe. M4 gives C₃ based on that SMA (i.e. positive if a is negative, and vice versa). N4 gives escape velocity from that altitude (which also appears in E8), and P5 and Q5 give the gate orbit for that destination, first as pure radius then as altitude after subtracting Kerbin's radius.

 

EDIT: I uploaded a slightly more explicitly useful version of the same spreadsheet here

Edited April 23, 2016 by Plusck

[Streetwind]

Slowly making my way through the (updated) sheet, already understanding what's what a fair bit better now...

 

1 hour ago, Plusck said:

J2 is Kerbin's orbital radius, which is also its SMA.

I assume that for a source body in an eccentric orbit, like Eeloo, I would always put the SMA here?

 

1 hour ago, Plusck said:

B6 and B7

What numbers am I entering here? From looking at C6 and C7, it would be... an altitude in kilometers? But why would you have a Pe1 of 200,000 kilometers, when Ap1 is only 75 kilometers? Do those names not mean "periapsis 1" and "apoapsis 1"? That's currently the thing that's tripping me up.

 

Edited April 23, 2016 by Streetwind

[Plusck]

28 minutes ago, Streetwind said:

Slowly making my way through the (updated) sheet, already understanding what's what a fair bit better now...

 

I assume that for a source body in an eccentric orbit, like Eeloo, I would always put the SMA here?

 

What numbers am I entering here? From looking at C6 and C7, it would be... an altitude in kilometers? But why would you have a Pe1 of 200,000 kilometers, when Ap1 is only 75 kilometers? Do those names not mean "periapsis 1" and "apoapsis 1"? That's currently the thing that's tripping me up.

 

Hey, a man can make a mistake in labelling his cells, no?

You're right - they are Ap and Pe respectively. They just ended up being the wrong way round for the graphing attempts. The yellow cells are altitude in km. The cell just to the right of them (named cells as "variabler1" and "variabler2") use the radius of the planet ("variableRadius") to get the actual radii in metres.

And the 200,000km was just an arbitrary number to get very near to "infinity" to double check my numbers. For KSP's purposes, that would be outside Kerbin's SOI. That also means that "escape velocity" is wrong for KSP's purposes, since it doesn't take the effect of reaching the edge of the SOI into account. It's correct for the "real world" Kerbin, however (I think...).

 

As for putting Eeloo's SMA in - I guess so. I really didn't intend this to be for public consumption - it just contains the main equations already in Excel format so you can play with them

Actually, now I think about it, it wouldn't work for the other planets. Kerbin's orbit is circular (I think), so it's orbital velocity is the same at all times and it's easy to get the SMA of the resulting orbit if you escape Kerbin directly prograde by adding the velocities and using the vice thingy equation. If you start from an elliptical orbit, you need to find the starting velocity at that point (which you can derive relatively easily using the various equations in that spreadsheet, basically copying the whole section B6:I6 and correcting all the cell references to be relative rather than absolute), then work from there.

And if you're starting from an arbitrary point in an elliptical orbit, you can then use that whole section A10:BL73 to work out where you are and what speed you'll be at and what angle you'll be flying at... more or less. It's the "more or less" bit that I hadn't yet sorted...

Edited April 23, 2016 by Plusck

[Streetwind]

14 minutes ago, Plusck said:

As for putting Eeloo's SMA in - I guess so. I really didn't intend this to be for public consumption - it just contains the main equations already in Excel format so you can play with them

Well, what I'm basically looking for is a spreadsheet where I input the stats for an origin planet and the stats for a destination planet, and it spits out the gate orbit around the origin planet for going to the destination.

Reason: I have this vision for setting up refueling stations in gate orbits to destinations find myself flying to often. But I won't be playing stock KSP, but rather New Horizons - a completely reordered solar system, where Kerbin doesn't even orbit the sun. Because I first need to go from Kerbin to Kerbin's parent planet's SoI before I can proceed to solar orbit, refueling stations in gate orbits would make double sense, because those stations would be located in orbit around Kerbin's parent. Then vessels could refill the fuel they expended to get out of Kerbin's SoI there. And vessels returning from interplanetary voyages could cheaply insert there and refuel before embarking on the final leg towards Kerbin.

So by having that spreadsheet, I could just input whatever numbers Kerbin's parent will turn out to have (once I'm done resizing the whole system to boot ) and the numbers of a destination, and I would know where to put a refueling station in that case.

Originally I intended to make this spreadsheet myself, but it's become clear that I need someone to hold my hand in the face of this very very scary math!

 

Edited April 23, 2016 by Streetwind

[Plusck]

29 minutes ago, Streetwind said:

Well, what I'm basically looking for is a spreadsheet where I input the stats for an origin planet and the stats for a destination planet, and it spits out the gate orbit around the origin planet for going to the destination.

Reason: I have this vision for setting up refueling stations in gate orbits to destinations find myself flying to often. But I won't be playing stock KSP, but rather New Horizons - a completely reordered solar system, where Kerbin doesn't even orbit the sun. Because I first need to go from Kerbin to Kerbin's parent planet's SoI before I can proceed to solar orbit, refueling stations in gate orbits would make double sense, because those stations would be located in orbit around Kerbin's parent. Then vessels could refill the fuel they expended to get out of Kerbin's SoI there. And vessels returning from interplanetary voyages could cheaply insert there and refuel before embarking on the final leg towards Kerbin.

OK, that makes a lot of sense.

I edited my previous post quite a bit, so there is maybe info there that you missed.

Basically, yes, you can do that with the equations you have here.

The problem with gate orbits in KSP is the SOI effect: because you get a cut-off at a certain distance out, the "real" equations will give results which don't correspond to the best orbits in KSP.

The other problem is that if you have a planet on an elliptic orbit, the gate orbit will be different depending on where it is in its orbit. If you decide to average and just take the SMA, that means the gate orbits will be best only in mid-orbit between Ap and Pe. And Ap and Pe are probably the best times to arrive or depart (like with Moho - you really want to arrive when Moho is at its Pe, because you're going faster than it anyway when you arrive and, therefore, you really don't want it to be going at its slowest when you get there... and your added velocity by going that bit further down the sun's gravity well is not as "different" as Moho's velocity is, if you see what I mean).

You shoud really repurpose the whole of the B6:I8 section (plus N6:N7) then, plus A1:B4.

You can forget about that local gravity bit (B1, G6 and G7).

Keep B2:B3 as a place to enter the variables for each planet. J2 should be moved to joint them. Put G2, H2 and I2 in a separate section since they are the relevant details for the sun.
For the moons, make duplicate pages where the planet's data is put in the place of the sun's data above, and the moon's data goes in B2, B3 and wherever you move J2 to.

Then you just need to determine the necessary C3 for each of your major transfers, and plug that in to get the gate orbit. To do that, you'll have to rearrange the equations I'm afraid.

[Plusck]

OK, well I couldn't resist fiddling with that spreadsheet. Then one thing led to another and I found myself making this.

It's a bit wasteful - but it was only once I'd copied everything across that I realised that a lookup function would have made it all far simpler.

Still, first page lets you input your planets (stock or New Horizons), and get all the gate orbits depending on whether you aim for the destination planet's Ap or Pe. Regarding that, the sheet does kinda prove the point about Moho: checking on the Alex Moon launch window page shows that yes, you can get to Moho with a 1500 m/s burn from LKO. The spreadsheet shows, however, that if you do that you will be faced with a 3800m/s insertion burn, because you are arriving at Moho's Ap.

Second page lets you use the data for the planets and add moon info. The example I put in was the Mun, going to Duna. Apparently the gate orbit is at about 100km above the Mun.

If I've made any mistakes, please correct them! Unfortunately I didn't have the patience to try and cater for moons on eccentric orbits. Nor did I give actual transfer burn values (which would have been useful) for each of those gate orbits.

[Streetwind]

On 25.4.2016 at 3:24 AM, Plusck said:

OK, well I couldn't resist fiddling with that spreadsheet. Then one thing led to another and I found myself making this.

Now that I finally have time and lack headaches again...

Cool sheet! That does indeed do what I envisioned. I have some questions though:

- You say it doesn't account for inclination. But wouldn't the gate orbit iself ideally be set up as coplanar with the target body's orbit? So you could burn from it without an inclination change. Or would doing that change the numbers because this spreadsheet assumes equatorial gate orbits?

- Regarding the SoI limit thing mentioned in J5, I understand what it means since it was dscussed further up in this thread, but... how large is the error that introduces actually? And is that error roughly the same in all cases, or does it differ wildly between different destinatio bodies (or between two origin bodies)? In Mr. Shifty's thread that you linked on page one, he's consistently posting altitudes slightly above the numbers your spreadsheet gives for SMA-to-SMA transfer. Do you think his calculation somehow accounted for this error?

- The "To Pe from AP" and vice-versa fields... that works out only if both planet's orbits have exactly the same argument of periapsis, right? Or, in the special case of Kerbin, when one of the two planets is in a perfectly circular orbit?

And a note: since gate orbits are circular, you can easily get the required transfer burn dV with alexmoon's transfer window planner. Though... that tool also defaults to changing the inclination during the transfer. Which works out well for starting from an equatorial gate orbit, but not for starting from a coplanar one. Hmm! Perhaps a request to TriggerAU, who manages the ingame adaption, for a "no inclination change" checkbox is in order...

[Plusck]

19 hours ago, Streetwind said:

Now that I finally have time and lack headaches again...

Cool sheet! That does indeed do what I envisioned. I have some questions though:

- You say it doesn't account for inclination. But wouldn't the gate orbit iself ideally be set up as coplanar with the target body's orbit? So you could burn from it without an inclination change. Or would doing that change the numbers because this spreadsheet assumes equatorial gate orbits?

- Regarding the SoI limit thing mentioned in J5, I understand what it means since it was dscussed further up in this thread, but... how large is the error that introduces actually? And is that error roughly the same in all cases, or does it differ wildly between different destinatio bodies (or between two origin bodies)? In Mr. Shifty's thread that you linked on page one, he's consistently posting altitudes slightly above the numbers your spreadsheet gives for SMA-to-SMA transfer. Do you think his calculation somehow accounted for this error?

- The "To Pe from AP" and vice-versa fields... that works out only if both planet's orbits have exactly the same argument of periapsis, right? Or, in the special case of Kerbin, when one of the two planets is in a perfectly circular orbit?

And a note: since gate orbits are circular, you can easily get the required transfer burn dV with alexmoon's transfer window planner. Though... that tool also defaults to changing the inclination during the transfer. Which works out well for starting from an equatorial gate orbit, but not for starting from a coplanar one. Hmm! Perhaps a request to TriggerAU, who manages the ingame adaption, for a "no inclination change" checkbox is in order...

- Coplanar doesn't work as soon as you cross an SOI boundary, because vectors. Specifically, if you're going prograde your ejection angle has to be greater than the target's inclinaton ([edit:] in all cases, the resulting triangle of vectors will always have a smaller angle than the ejection angle) and if you're ejecting retrograde it has to have the opposite angle, again at a greater angle (unless the ejection velocity is so high you end up halving your solar orbital velocity).
Taking Moho as an example (Moho is really a great example for lots of stuff): in order to get the absolute cheapest overall trip to Moho, you have to launch at Kerbin/Moho relative AN/DN, and you have to launch into an inclined orbit from Kerbin. However, the absolute best inclined orbit is the opposite inclination as Moho (7°), and then some. From the spreadsheet we get a V_infinity of 2900 m/s on leaving Kerbin's SOI to reach Moho at Pe. Assuming AN/DN coincides with Pe (which it doesn't, AFAIK, but let's just assume...) our resulting orbital velocity will be Kerbin's (9284m/s) at 0°, plus V_infinity (2900m/s) at, say, 183° (compared to the plane of the orbit, I'm assuming a perfectly retrograde ejection under Kerbin's orbital plane), i.e. -7°. From the law of cosines (c² = a² = b² - 2ab cos C) our resultant velocity in solar orbit is 6415m/s. Using the law of sines (sin A/a = sin C/c), our angle to the orbital plane is actually only 3.2°... and our Pe radius from Kerbin's orbit with that velocity is (from a = GM/(2GM/r₁-v²), and r₂=2a-r₁) = 4,265Mm. So by matching the negative inclination of where we were going, we end up with a smaller inclination and an insufficient burn to reach Moho's Pe (4,210Mm).

Therefore, to get the right inclination of your orbit around a planet or moon to match the inclination of another planet or moon's orbit, you would have to reverse these equations: start with the known resulting vector angle and magnitude (i.e. 7° and (9284-2900)=6384m/s for Moho), and going back to the law of cosines this gives a required V_infinity of 3048m/s and, from the law of sines, a burn angle of 14.8° normal or anti-normal from Kerbin.

And bear in mind that this inclined gate orbit will be less useful at any time other than the one time every year that it matches your destination perfectly. On the plus side, though, if you put a refuelling station in Kerbin orbit at 14.8°, you'd always have a target orbit to launch into if you're doing a lot of Moho launches.

- The SOI effect is variable, but significant. Generally I think KSP's planets have a reasonable limit for their SOI, but the moons don't (especially within the Jool system, for obvious reasons). This always means that you'll need a slightly lower C3, and therefore the gate orbit will be higher. So I guess the spreadsheet should be completed by a calculation of the difference... but I'm at a bit of a loss to know how to do that easily.

- For the "to Ap from Pe" and vice versa, yes, this is only going to be true in that case. The aim was not so much to give a real figure, but to give a best-case and worse-case scenario, to set the bounds of possible error around that "to SMA from SMA" figure. It was also intended to dispel the idea (that one might have intuitively) that the lowest transfer burn gives the best transfer: in all cases, it doesn't unless you can get a gravity assist or aerobrake to capture.

Edited April 27, 2016 by Plusck

[Streetwind]

Yeah, I soundly messed that thing up in my head with the transfer ellipse.

But thanks again for your help, have some likes poured onto you!