Stupid question regarding orbital periods

[photogineer]

So, in my quest to do less eyeballing and more mathy stuff I thought about trying to figure out how to calculate and predict orbital intercepts for docking instead of just yanking at nodes until things line up. I've hit a weird stumbling block, so I am hoping more experienced rocketeers can help me out. Stupid question below

I am trying to understand how to calculate the period of an orbit, elliptical or circular, and in my searching I came across this:

http://en.wikipedia.org/wiki/Orbital_period#Small_body_orbiting_a_central_body

That is simple enough, and it makes sense, but I'm a little perplexed because the article says the formula is for elliptical or circular orbits and only depends on the semi-major axis. Does that mean in a situation where you have a circular 250km orbit, and a 250kmx150km elliptical orbit, they have the same period? I know the velocity will vary over that orbit, but it seems like they shouldn't be the same, am I reading that correctly?

Matt

Edited November 20, 2013 by photogineer

[Meithan]

That formula's correct (and by the way, it's pretty much Kepler's Third Law). The period of a small body in orbit about a central body of mass M depends directly on the length of the semi-major axis a of the small body's orbit:

As you guessed, that means it's possible to have a circular orbit that has the same period as an elliptic one, although your example wasn't quite correct (remember that semi-major axis is just the average of the periapsis and apoapsis distances). A circular orbit at 250 km altitude above Kerbin's surface would have the same period (43m 44s) as a 150 km x 350 km elliptic orbit (since its semi-major axis is the same). Notice that for that to work, the periapsis must be lower than 250 km while the apoapsis higher than that. If the elliptic orbit is completely below or above the circular one, they will have different periods.

[phunk]

They won't be the same. The semi-major axis is not measured in reference to the planet, it's measured from the center of the orbit.

Kerbin's radius is 600km, that means a 250km circular orbit has a semi-major axis of 850km. A 250km x 150km orbit will have a semi-major axis of 800km I believe.

[maltesh]

The two orbits you have mentioned do not have the same semi-major axis. KSP only gives you the altitude above the surface directly, and you have to remember that Kerbin has a radius of 600km.

Semimajor axis is half the longest distance across the ellipse. For a circular orbit, it's the radius, for an elliptical orbit, it's half the linear distance between periapse and apoapse.

A circular orbit whose altitude is 250 km over Kerbin's surface has a radius of 850 km.

An elliptical orbit over Kerbin whose periapse is at 150 km altitude, and whose apoapse is 350 km altitude also has a semimajor axis of 850 km.

In the case of the two orbits I just mentioned, yes, they would have the exact same period.

Another interesting point ; For any two orbits around the same body who have the same semimajor axis, at any distance from the central body that the two orbits share, orbital speed will be the same. In the above example, an object in the elliptical orbit would have the same speed as an object in the circular orbit when it crosses the 250 km altitude line.

[Meithan]

The semi-major axis is not measured in reference to the planet, it's measured from the center of the orbit.

While that's true, if the average of the altitudes is the same, then so is the average of the distances to the center of the planet. That is, we can determine whether two orbits have the same period by comparing their periapsis/apoapsis altitudes. I think it was just a miscalculation on his part.

Edited November 20, 2013 by Meithan

[photogineer]

Aha! Alright, that all makes sense, and I see my error, I was treating the radius of the ellipse at the longer end as my semi-major axis, that was an error on my part, and it all now makes sense to me. Thanks all.

Matt

[phunk]

I was treating the radius of the ellipse at the longer end as my semi-major axis, that was an error on my part

That's not an error, that is the semi-major axis. I think your error is in calculating that radius.

ETA: As Meithan mentioned, it's the average of the apoapsis and periapsis, plus the radius of the planet.

[Kasuha]

If you don't want to use math and are just interested in the period, check what's written at AP and PE. Subtract the two times and multiply the result by two. For instance the orbit in the picture has period (33:20 - 16:23) x 2 = 33:54.

[sanoj688]

So guys, what if I wanted to increase my orbital period with 25% while keeping my periapsis the same? How to find out which apoapsis you need to reach?

I want to get into a geosync orbit, then increase my orbital period with 25% and launch 4 probes.. then wait 1 orbit, circularize 1 probe and repeat the process for the other 3 probes. That gives me 4 probes (comsats) in geosync orbit, equally spaced across the orbital trajectory..

I tried googling and using keplers third law but couldn't find a good online calculator for it..

[Meithan]

You can use the expression for the orbital period I gave before and the fact that the semi-major axis is the average of periapsis and apoapsis to solve for the apoapsis if the period and periapsis are fixed.

Rearranging the aforementioned expression for orbital period, you can write

Then, use the fact that to solve for the apoapsis distance. You'll arrive at:

where T is the orbital period that you want and rpe is the periapsis distance.

By the way, you might want to look at this thread, in which I showed an arrangement of keosynchronous satellites very similar to what you're planning to do.

Edit: and if you want to know more about the details, check out my mission log.

Edited November 21, 2013 by Meithan

[Dogface]

Interactive illustrated interplanetary guide and calculator for KSP

http://ksp.olex.biz/

[Meithan]

So guys, what if I wanted to increase my orbital period with 25% while keeping my periapsis the same? How to find out which apoapsis you need to reach?

I want to get into a geosync orbit, then increase my orbital period with 25% and launch 4 probes.. then wait 1 orbit, circularize 1 probe and repeat the process for the other 3 probes. That gives me 4 probes (comsats) in geosync orbit, equally spaced across the orbital trajectory..

I tried googling and using keplers third law but couldn't find a good online calculator for it..

I crunched the numbers for you. In fact, it's cheaper that you shoot for a "delivery orbit" with a period that is 25% smaller instead of larger. An orbit with a period of 4.5 hours (16200 seconds) that has its apoapsis at the keosynchronous altitude (2868.6 km) would have a periapsis altitude of 1658.0 km.

The process is then as you described: you release a probe at apoapsis, have it circularize its orbit, then wait one orbit (4.5 hours) and when you're back at apoapsis you release the second probe, circularize its orbit, and so on. You'll get four nearly equally spaced probes.

[KerbMav]

Concentrate more on giving them the correct orbital period than nailing AP and PE, a few kilometers will have no negative effect and even a period <>6h will give you a working sat-network if they are all off by the same numbers.

[Meithan]

Concentrate more on giving them the correct orbital period than nailing AP and PE, a few kilometers will have no negative effect and even a period <>6h will give you a working sat-network if they are all off by the same numbers.

Well, that's easy: he'd want a delivery orbit with a period as close to 4.5 hours as possible. I just think it's easier to read the periapsis/apoapsis distances directly from the game than figuring out the period from the "time to periapsis" and "time to apoapsis".

[KerbMav]

Well, that's easy: he'd want a delivery orbit with a period as close to 4.5 hours as possible. I just think it's easier to read the periapsis/apoapsis distances directly from the game than figuring out the period from the "time to periapsis" and "time to apoapsis".

It was meant as an addendum to establishing sync-orbits - but you are right, I have gotten used to the readouts of KER.