How to calculate phase angle for hohmann transfer?

[sodra]

Lets say I have a space station at a 100km orbit, and a shuttle at 200km orbit.

How do I calculate the phase angle for doing the first burn (with the shuttle, to intercept the station) with the least delta-v?

EDIT: Please don't refer me to the Scott Manley video. He is an amazing rocketeer, but I can't find the part in the video where he explains it.

Edited February 7, 2014 by sodra
Question Answered

[TheDarkStar]

Using maneuver nodes, you don't need to calculate it. Set your target, set the node to just barely hit the target orbit, and then move the node around by dragging it until you get a close encounter. After that, just circularize.

[sodra]

Using maneuver nodes, you don't need to calculate it. Set your target, set the node to just barely hit the target orbit, and then move the node around by dragging it until you get a close encounter. After that, just circularize.

I know I could do it with maneuver nodes, but doing it by hand it more fun to me.

Plus I can do a more accurate rendezvous.

[Specialist290]

As with many complex calculations, it's easiest to get a handle on if you break it down into its component tasks. We'll assume for the sake of convenience that the origin and destination orbits are circular, or close enough to circular for required adjustments to be negligible. For the specific equations, you may want to refer to Robert Braenuig's page on orbital mechanics. I'll throw in an additional disclaimer that math was never exactly my strong suit in school, so don't expect me to be able to explain these equations inside-out myself.

First, you'll want to calculate the parameters of your transfer orbit, with one apsis at the origin altitude and the other at the destination altitude. Once you've done that, take half of that orbit's period, since that will be the amount of time it will take for your craft to travel from one apsis to the other. With that, you'll want to figure out the length of the arc the target craft travels in its orbit in the same amount of time. You can do this by dividing the half-period of the transfer orbit by the period of the destination orbit; the ratio between the two will be the fraction of the orbit's radius that the target craft travels in that time. Convert that ratio to degrees, and you'll have the phase angle you want.

Hope this helps

EDIT: One of my peers has also let me know that he's posted this handy equation that boils all of what I've said down into a single formula.

Edited February 7, 2014 by Specialist290

[Geschosskopf]

I know I could do it with maneuver nodes, but doing it by hand it more fun to me.

Plus I can do a more accurate rendezvous.

You won't notice much if any accuracy improvement. First off, there's nothing at all in the stock game that measures the angle between planets so if all you know is the phase angle, you'll just have to eyeball it in the map view, which isn't conducive to accuracy. To make the phase angle useful, you have to know what day the planets will line up, so you can warp time forward however many months until that day arrives. OK, that's as good as you can get there.

But then there's also the question of ejection angle, meaning where on your starting orbit you need to place the maneuver node. While you can calculate this angle as well, there is still absolutely nothing in the stock game that measures this angle, so you just have to eyeball it, which again destroys accuracy.

The only way to avoid the inaccuracies of eyeballing angles is to use mods. Protractor measures angles so you can use it for both phase and ejection angles. Kerbal Alarm Clock knows what days the phases angles for various transfers come up and will stop the clock at them but it's only good for the phase angle. MechJeb will plunk down a maneuver node for you at the correct ejection angle on the proper day.

[EtherDragon]

I really need to get to putting together my big video series on Orbital Mechanics - this similar question has come up a lot lately...

Anyway, some call it "Phase Angle" and such, but what you are simply looking for is the portion of the Target's Orbit that will be completed when you are on you Hohmann Transfer Orbit. Well, there's a simple equation I worked up for exactly that:

p = 1 / (2*sqrt (d^3 / h^3))

P gives you the portion of the target's orbit given that

d is the Destinations Semi-Major Axis and

h is your Hohmann Transfer Semi-Major Axis

Not sure what your Hohmann Transfer Semi-Major Axis is? If you look at how to calculate a normal Semi-Major Axis (p+a)/2 where 'p' is your Periapsis and 'a' is your apoapsis you just plug in your two hypothetical values: 'p' is your current orbit height, and 'a' is the height you wish to reach. Thus:

h = (s+d)/2

p = 1 / (2*sqrt (d^3 / h^3))

where

s = your starting semi-major axis

d = your destination semi-major axis

h = your transfer semi-major axis

giving

p = the portion of the target orbit completed while you coast.

Specifically, for a 100km parking orbit, and a destination orbit of 200km:

h = (700+800)/2 = 750

p = 1 / (2*sqrt (800^3 / 750^3)) = ~0.454 orbits, or not quite half an orbit.

(Now I'm off to KSP to test the results... be right back...)

EDIT: Yep - I was correct. I was able to plan an encounter with the target - departure was set for when I (at 100km altitude) was a smidge behind the target (at 200km altitude).

Wait, why did I put 700 and 800 in for the semi-major axis? You have to add Kerbin's Radius to your altitudes to get the SMA. Anyway, good luck!

EDIT EDIT: FYI this same equation can be used to find optimal interplanetary angles as well...

Edited February 7, 2014 by EtherDragon

[sodra]

I really need to get to putting together my big video series on Orbital Mechanics - this similar question has come up a lot lately...

Anyway, some call it "Phase Angle" and such, but what you are simply looking for is the portion of the Target's Orbit that will be completed when you are on you Hohmann Transfer Orbit. Well, there's a simple equation I worked up for exactly that:

p = 1 / (2*sqrt (d^3 / h^3))

P gives you the portion of the target's orbit given that

d is the Destinations Semi-Major Axis and

h is your Hohmann Transfer Semi-Major Axis

Not sure what your Hohmann Transfer Semi-Major Axis is? If you look at how to calculate a normal Semi-Major Axis (p+a)/2 where 'p' is your Periapsis and 'a' is your apoapsis you just plug in your two hypothetical values: 'p' is your current orbit height, and 'a' is the height you wish to reach. Thus:

h = (s+d)/2

p = 1 / (2*sqrt (d^3 / h^3))

where

s = your starting semi-major axis

d = your destination semi-major axis

h = your transfer semi-major axis

giving

p = the portion of the target orbit completed while you coast.

Specifically, for a 100km parking orbit, and a destination orbit of 200km:

h = (700+800)/2 = 750

p = 1 / (2*sqrt (800^3 / 750^3)) = ~0.454 orbits, or not quite half an orbit.

(Now I'm off to KSP to test the results... be right back...)

EDIT: Yep - I was correct. I was able to plan an encounter with the target - departure was set for when I (at 100km altitude) was a smidge behind the target (at 200km altitude).

Wait, why did I put 700 and 800 in for the semi-major axis? You have to add Kerbin's Radius to your altitudes to get the SMA. Anyway, good luck!

EDIT EDIT: FYI this same equation can be used to find optimal interplanetary angles as well...

Thank you, this is exactly what I needed.

I'm using MechJeb to find the information I need, but I'm going to do the burns myself.

Seems more fun that way.

[AlamoVampire]

hey sodra, the quicker way, where it does not involve cranking the equations manually: ksp.olex.biz OR use mechjeb

[sodra]

hey sodra, the quicker way, where it does not involve cranking the equations manually: ksp.olex.biz OR use mechjeb

That calculator is only for interplanetary transfers.

I do use MechJeb, but I would like to learn how to do a rendezvous by hand, in case I miraculously get trapped inside a Gemini Capsule while in orbit one day, and I need to rendezvous with the ISS and EVA out.

[Max Grant]

I usually use the edge of a book when working these things out. If my book's edge crosses into the orbit of my origination, it needs to be a bit further along.