Maths (very very frightening)

[GarrisonChisholm]

Out of curiosity, which return to Kerbin would be cheaper; from a 60km orbit around Duna, or a 10km orbit around Ike?

 

[added]

I am also rather tickled that this thread has fostered such a prolonged mathematics discussion, despite the fact that I would find it very very frightening if I were compelled to attempt any of these resolutions!

Let one and all feel free to interject their miscellaneous math musings below and here-in.

Edited April 26, 2016 by GarrisonChisholm

[HoloYolo]

Ike. It's orbital velocity around Duna + Your Orbital Velocity + Duna's orbital velocity around the Sun adds up to a cheaper ejection burn.

[Enorats]

Ike would be cheaper, but likely trickier to time right. It might not be at the right place during the transfer window. Also, getting to that orbit around Ike from the low Duna orbit would likely make it more expensive. Duna is a little larger (but only a little), so the Oberth effect will be a little stronger with a low Duna orbit.

[Kerbart]

Assuming you don't go back to low Duna orbit first. “But that's what the delta-V map tells me to do...”

[Guest]

2 minutes ago, Enorats said:

Ike would be cheaper, but likely trickier to time right. It might not be at the right place during the transfer window.

Surely your transfer window can accommodate a 24 hour margin of error...

[cantab]

Yeah, that would only be an issue if leaving from a highly-inclined orbit, round either Duna or Ike.

[Renegrade]

45 minutes ago, HoloYolo said:

Ike. It's orbital velocity around Duna + Your Orbital Velocity + Duna's orbital velocity around the Sun adds up to a cheaper ejection burn.

Ike's orbital velocity is only ~314m/s though.   A 10km Ike orbit orbit is about 365.18m/s - that's less than a 60km Duna orbit (890.54).  In either case Duna's orbital velocity is the same of course (it's like 7500m/s or thereabouts).

Of course, you aren't as deep in Duna's gravity well up at Ike (~2.8 Mm) - escape velocity from that point is ~440m/s (from Duna's SOI) vs 1259.41 for a 60km Duna orbit.

...too sleepy to figure it out... ugh.  I'll look at it again later.

BTW, Ignoring Oberth, the deep-space transfer from Duna's orbit to Kerbin's orbit is roughly 1760m/s if that helps anybody's calculations (first impulse is about 833m/s).

[GarrisonChisholm]

I appreciate the input!  I had test-flown my Duna craft unmanned, but now as bean-counters do it was decided to add Scientists (tourists) to the mission to increase profitability, and one of them needs to Orbit Ike.  So, I *should* have plenty of dV to land at Duna, return to orbit, & then orbit Ike before burning for Kerbin, but I just wanted to be sure my initial thought was correct; that there would be some small savings from departing from a much lower orbit around Ike.

That makes me feel more comfortable with the planning. 

I have ~ a year before my launch window, plenty of time to get the ducks in a row.  - I'm also ensuring that *all* other ongoing flights are concluded, so Duna will be the only thing to focus on for 2 1/2 years.

[Snark]

Gameplay question moved to Gameplay Questions.

[Nich]

1 hour ago, regex said:

Surely your transfer window can accommodate a 24 hour margin of error...

or even a week of leeway is fine.

 

A long time ago someone made a spreadsheet of all the moons transfer is cheaper from and cheaper to burn retro grade to the central gravity well and then burn transfer from there.  Mun is always cheaper to burn out directly, Minimus is always cheaper to burn to kerbin then burn out.  Because ike is so large and low in Dunas gravity well I suspect it is cheaper to burn out from ike.  This is exacerbated by the fact the duna doesn't have much of a gravity well to take advantage of.

[Superfluous J]

Why do the math? HyperEdit ships into the appropriate orbits (or load an old save with ships already there if you for some reason consider it "cheating") and draw maneuver nodes.

[GoSlash27]

Before I give the answer, let me outline how the math works. It's much less frightening than you think.

Step 1: Figure out the Hohmann transfer from one body to the other.

This uses the uniform circular motion and vis-viva equations. They're just plug 'n' play. Subtract Duna's orbital velocity about the sun to get the excess velocity; Vxs.

Step 2: Figure out your orbital velocity about Duna at your current altitude.

Uniform circular motion.

Step 3: Figure out escape velocity from Duna.

This is always your orbital velocity times the square root of 2. Vorb*sqrt(2)= Vesc.

Step 4: Figure out your transfer velocity.

Vtx= sqrt(Vesc^2+Vxs^2)

Step 5: Subtract your orbital velocity from the transfer velocity.

DV= Vtx-Vorb.

 

 For a transfer from low Ike orbit, you would calculate a theoretical burn from Ike's orbital radius and treat it as Vxs, and repeat steps 3 through 5 for Ike's SoI.

That's really all there is to it. It's tedious, but not difficult. Or at least it's tedious until you set up a spreadsheet to do it for you.

Best,
-Slashy

 

Edited April 7, 2016 by GoSlash27

[GoSlash27]

So for the answer...

A Hohmann transfer from 60km Duna orbit to Kerbin is 615.6 m/sec

A Hohmann transfer from 10km Ike orbit to Kerbin is 446.6 m/sec.

You would save 169 m/sec by transferring direct from Ike.

 

Best,
-Slashy

 

 

 

Edited April 7, 2016 by GoSlash27

[GarrisonChisholm]

Much obliged for your expertise Slashy, gracias.

Despite my two favorite hobbies (KSP & Baseball) being math heavy, math ruined my life from ages 11-16, so whenever I see "X=..." my eyes just roll back in my head and I start crying blood & spitting tears.  Why didn't Khan Academy exist in 1981?...

Thank you.

[GoSlash27]

GarrisonChisolm,

 You're welcome and de nada.

 I never got into math much when I was in school either, but I found out later on that I actually enjoyed it when there wasn't so much pressure.
 If you hang around KSP, you will eventually get very good at math. It's a gateway drug

Best,
-Slashy

[Just Jim]

28 minutes ago, GoSlash27 said:

If you hang around KSP, you will eventually get very good at math. It's a gateway drug

 

Man, if we could just get more people hooked, this world would be such a better place..... 

[Andem]

1 hour ago, Just Jim said:

 

Man, if we could just get more people hooked, this world would be such a better place..... 

Undoubtedly!

[Kosmognome]

No post here took the Oberth effect into account. So while you might save 169m/s by transferring from Ike, the question is wether the Oberth effect gives you enough free delta-v to make up for it.

 

[Plusck]

A thread I keep plugging that can also help:

From this you can see that the most efficient altitude to burn from Duna to Kerbin is at 580km, therefore a 60km orbit is not the best place to burn from.

As always, raising your orbit before a transfer burn is guaranteed to be a waste of energy. However, lowering it may or may not help depending how far you are from that "most efficient" orbit.

Maybe @GoSlash27 could plug a 580km Duna orbit into that spreadsheet he hinted at? Or I could be less lazy and try to do it myself...

 

[FancyMouse]

32 minutes ago, Kosmognome said:

No post here took the Oberth effect into account. So while you might save 169m/s by transferring from Ike, the question is wether the Oberth effect gives you enough free delta-v to make up for it.

 

The 169 number does. It has some approximations (e.g. the sqrt(2) factor is literally ignoring the gravitational potential energy at SoI boundary) but won't change the result too much. The step that it pulls the velocity down to your current orbit around Ike and then subtract is Oberth effect.

[GoSlash27]

2 hours ago, Plusck said:

A thread I keep plugging that can also help:

From this you can see that the most efficient altitude to burn from Duna to Kerbin is at 580km, therefore a 60km orbit is not the best place to burn from.

As always, raising your orbit before a transfer burn is guaranteed to be a waste of energy. However, lowering it may or may not help depending how far you are from that "most efficient" orbit.

Maybe @GoSlash27 could plug a 580km Duna orbit into that spreadsheet he hinted at? Or I could be less lazy and try to do it myself...

 

Plusck,

 Sure thing. DV from 580km Duna orbit is 584.1 m/sec, and it's definitely the minimum for a direct departure from Duna orbit. And you're absolutely correct; there is a conflict between Oberth and the gravity well. Too far down increases your gains from Oberth, but those gains are outstripped by gravity losses. There is a "happy spot" for each Hohmann transfer.

2 hours ago, Kosmognome said:

No post here took the Oberth effect into account. So while you might save 169m/s by transferring from Ike, the question is wether the Oberth effect gives you enough free delta-v to make up for it.

Kosmognome,

 I can assure you, Mr. Oberth is accounted for

Best,
-Slashy

 

[GoSlash27]

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

[Gojira1000]

7 minutes 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

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

[Vegetal]

2 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

Hey that's very interesting! And useful!

A question from an uninitiated: r is radius, u is what? And excess velocity is what exactly? The delta V of your escape burn? Thx.

[GoSlash27]

1 hour ago, Vegetal said:

Hey that's very interesting! And useful!

A question from an uninitiated: r is radius, u is what? And excess velocity is what exactly? The delta V of your escape burn? Thx.

Vegetal,

 u would be "mu"; the parent body's mass times the universal gravitational constant.

Excess velocity... If you figure out the DV required for a Hohmann transfer from your current planet's orbit about the sun to your destination's orbit about the sun and completely ignore the planets themselves, that would be the excess velocity. 

Best,
-Slashy