Oberth Effect (calculation of benefit)

[Anglave]

I'm no math whiz, but I'm interested in a better fundamental understanding of the Oberth Effect, so I tackled some equations.

What is the proper way to calculate the ÃŽâ€v gained because of the Oberth Effect?

Let's use a concrete example. I'm in an 80km circular equatorial orbit of Kerbin. I want to plot a Jool intercept.

I'll assume the ÃŽâ€v required to go from LKO to Jool encounter is ~1920m/s

I'm also simplifying things a bit, assuming instantaneous burns and no plane changes, etc.

UPDATE -- THE REST OF THIS POST IS JUST PLAIN WRONG. Might as well not waste your time, just skip ahead to the answers.

Doing my best to follow the equations and examples provided by nyrath on this page:

First we calculate Vesc using:

Vesc = sqrt((2 * G * M) / r)

where:

G = gravitational constant (of the Kerbal universe)

M = mass you're orbiting

r = radius of your orbit

and μ = G*M. I'm using this because μ is provided (as Standard Gravitational Parameter) on the KSP Wiki entry for Kerbin. Kerbin's radius (600km) is also provided there.

For Kerbin:

μ = 3,531,600,000,000 m3/s2

r = 600,000m + 80,000m = 680,000m

sqrt(7,063,200,000,000 / 680,000) = 3222.89m/s or Vesc = 3.22 km/s

-------

Then we find ÃŽâ€v, or the total change in m/s required at periapsis to achieve the desired final velocity

using:

ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2)

where:

ÃŽâ€v = amount of burn at periapsis

Vf = final velocity target [1920 m/s (Jool Intercept) + 2280m/s (your current velocity) = 4,200m/s]

Vh = initial velocity before burn = 2,280 m/s [orbital speed at 80km]

and Vesc = 3,223 m/s

[check on my assumptions: Yasmy's orbital calculations came up with ÃŽâ€v = 4224 - 2246 = 1978 m/s. Mine came up with 4200 - 2280 = 1920. These numbers seem to more or less agree.]

So we simplify and find:

sqrt(17640000 + 10387020) - sqrt(5198400 + 10387020)

5294 - 3948 = ÃŽâ€v 1346 m/s

So, to the best of my ability, I think to get the desired increase of 1,920 m/s, it actually costs us only 1,346 m/s because of the Oberth Effect.

A savings of 574 m/s over burning with a solar velocity of 0.

---------------

This (if I've done my math correctly, which is seriously in doubt) answers the question "how much ÃŽâ€v do I effectively get 'for free' when burning from LKO to Jool intercept". If, however, you were to burn at some other point, say if you were performing a powered gravity assist past Mun on your way out of Kerbin's SoI, you would gain less 'free' ÃŽâ€v because your velocity would be lower.

-----------

How do I reconcile this with Yasmy's well articulated point that the proper way to measure the value of the Oberth Effect is to find the difference between

1) A single LKO to Jool burn, which we've taken to be 1920 m/s, and

2) Two burns, one from LKO to just barely Kerbin escape; and a second from just outside Kerbin's SoI (at periapsis of your solar orbit) to Jool encounter. The combination of which is just less than 3727.

Giving an answer of something close to 1740 m/s saved. due to the Oberth Effect.

Which answer is correct, and why?

--Update-- trying to answer my own question.

What happens if I try to explicitly use velocity in the solar frame, rather than the Kerbin frame? Is that my error above?

ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2)

where:

ÃŽâ€v = amount of burn at periapsis (solve for this)

Vf = final velocity target: 12.01km/s [velocity at Kerbin's orbital radius of a Kerbin to Jool transfer orbit. (from Yasmy's example)]

Vh = initial velocity before burn [Kerbin's orbital velocity 9.2km/s, plus your Kerbin orbital velocity 2.28km/s = 11.48km/s]

Vesc = 3,223 m/s

simplifying: sqrt(144240100 + 10387729) - sqrt(131790400 + 10387729)

12053 - 11924 = ÃŽâ€v = 129 m/s.

Meaning we'd only need to expend 129 m/s at periapsis to achieve the desired 530 m/s change.

A savings of 401 m/s. That's not closer to Yasmy's answer!

----

Something above can't be correct. I know I need more than 530 m/s change to reach Jool from LKO. Obviously I've done something wrong.

What if I just force the difference between Vf and Vh to be 1920 m/s? That would mean if Vh is 12,010 m/s, Vh must be 10,090 m/s

simplifying: sqrt(144240100 + 10387729) - sqrt(101808100 + 10387729)

12053 - 10592 = ÃŽâ€v = 1461 m/s.

Meaning we'd only need to expend 1461 m/s at periapsis to achieve the desired 1920 m/s change.

A savings of 459 m/s because of the Oberth Effect.

Grrr, now I have no idea where I've gone wrong, and none of my answers are any closer to agreeing with Yasmy's.

---- Another Update ----

I just realized I'm still using Vesc for Kerbin, not Kerbol. Maybe that's where I've gone wrong yet again?

Recalculate Vesc:

Vesc = sqrt((2 * G * M) / r)

where:

G = gravitational constant (of the Kerbal universe)

M = mass you're orbiting

r = radius of your orbit

μ = G*M.

μ = 1.1723328×10^18 m^3/s^2

r = 261 600 000 (Kerbol's radius) + 13 599 840 256 (Kerbin's orbital distance) = 13,861,440,256 m

Simplifying: sqrt(2,344,665,600,000,001,165 / 261,600,000) = sqrt(8962788991) = Vesc = 94,672

Which we then plug in:

sqrt(144240100 + 8962787584) - sqrt(101808100 + 8962787584)

95431 - 95208 = 151 m/s

Telling us that we need only expend 151 m/s to achieve the desired 1920 m/s change. A savings of 1769 m/s

Well, at least that's closer to Yasmy's answer. Can anyone tell me if I've finally solved it, or simply plugged in enough random numbers to get lucky? Spoiler -- it was the second thing.

Also (and finally), I know from experience that it takes considerably more than 151 m/s ÃŽâ€v to transit from LKO to Jool encounter. Why is that?

---FINAL UPDATE TO OP--- Snark pointed out the error of my ways in post #2. My basic assumption was incorrect, and therefore the math above is nonsensical.

Edited July 9, 2015 by Anglave
nonsensical

[Snark]

Let's use a concrete example. I'm in an 80km circular equatorial orbit of Kerbin. I want to plot a Jool intercept.

I'll assume the ÃŽâ€v required to go from LKO to Jool encounter is ~1920m/s

Yes, that's correct. You can verify this by using http://ksp.olex.biz to find the burn needed to go from Kerbin to Jool from an initial parking orbit of 80 km. It says "1911 m/s", which is close enough to the 1920 number. I've used this tool for lots of interplanetary transfers, so there's plenty of experimental verification that what it's doing is correct.

1920 m/s is the burn you need to do to get from Kerbin to Jool, taking maximum advantage of Oberth effect. So if you want to know "how much benefit do I get from Oberth", you'd need to work out "how much more than that would I need if I didn't use Oberth effect". That's what Yasmy's doing, and I believe he's doing it correctly.

Let's consider what that 1920 m/s burn buys you:

- It buys you escape from Kerbin's gravity well.

- It does so with enough energy left over to kick your solar apoapsis up to Jool's orbit.

First we calculate Vesc using:

Vesc = sqrt((2 * G * M) / r)

[...math...]

Vesc = 3.22 km/s

Yes, your math looks about right, but bear in mind that you have to be careful about actually using that number for anything. Escape velocity is to escape to "infinity" in real physics, which doesn't apply in KSP because of patched conics. Kerbin's gravity totally ceases at the boundary of its SoI, so really what you should be calculating is "how much energy do I need to be able to get to the boundary of the SoI". However, the Vesc will be pretty close to that, so it's good enough as a first-level approximation.

Then we find ÃŽâ€v, or the total change in m/s required at periapsis to achieve the desired final velocity

I'm confused. Why would you do any calculation here? The answer is 1920 m/s. "How much dV do I need to get 1920 m/s dV" is a question that has already answered itself. Bear in mind that 1920 isn't any kind of "desired final velocity" (for example, it's not the velocity you exit Kerbin's SoI with). It's simply the extra velocity above-and-beyond LKO orbital velocity that is needed to get to Jool. It has nothing to do with the Kerbin-relative velocity that you have when you pass the SoI boundary.

using:

ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2)

where:

ÃŽâ€v = amount of burn at periapsis

Vf = final velocity target [1920 m/s (Jool Intercept) + 2280m/s (your current velocity) = 4,200m/s]

Vh = initial velocity before burn = 2,280 m/s [orbital speed at 80km]

I'm not sure where you're getting the equation that I highlighted in red, above. The "ÃŽâ€" in "ÃŽâ€v" simply means "amount of change."

The correct equation there doesn't involve any square roots or whatever, it's just

ÃŽâ€v = Vf - Vh

...i.e. 1920 m/s.

Also not sure why you're adding in Vesc to your equations above.

So the bottom line: Yasmy's correct, due to explanation there. I can tell that you're making a mistaken assumption somewhere (because you get radically different numbers) ... but it's hard to pinpoint exactly what the mistaken assumption is, because you present the highlighted equation without any additional information-- if you could describe what you're trying to calculate and why (i.e. where you get that equation from, or how you derived it), that would be helpful.

- - - Updated - - -

So here's how to set up the math:

R0 = radius of Kerbin LKO, i.e. 680,000 m

R1 = radius of Kerbin SoI, i.e. 84,159,286 m

Rk = radius of Kerbin's orbit around the sun, i.e. 13,599,840,256 m

Rj = radius of Jool's orbit around the sun, i.e. 68,773,560,320 m (I'm using semi-major axis as an approximation, since Jool's orbit is slightly elliptical)

μk = G*M for Kerbin, i.e. 3.5316000×10¹² m³/s²

μs = G*M for sun, i.e. 1.1723328×10¹⁸ m³/s²

So with that in mind, here's how you would set up the math:

1. Imagine that Kerbin doesn't exist. You're orbiting the sun at Rk, and you want to boost your solar apoapsis up to Rj. How much ÃŽâ€v do you need? Call this Vt (for "transfer). You calculate this as follows:

- work out solar orbital velocity for circular solar orbit at Rs

- work out the periapsis velocity for an elliptical orbit with periapsis at Rs and apoapsis at Rj

- Vt = the difference between these two

2. This Vt is how much velocity you need to exit Kerbin's SoI with.

3. For the non-Oberth (i.e. wasteful, foolish) way to get to Jool: Work out how much ÃŽâ€v you need to get from LKO to the boundary of Kerbin's SoI. Call this Vesc. It's the difference between "orbital velocity at R0" and "periapsis velocity for elliptical orbit with periapsis at R0 and apoapsis at R1". This will be pretty close to the theoretical Vesc that ignores patched conics. Add Vt + Vesc, this gives you ÃŽâ€v needed for a "non-Oberth" transfer.

4. Now let's try bringing in Oberth. "I'm traveling at V1 at R0, and I want to be going Vt at R1, how much faster than V1 do I need to go?" This is the number that will work out to about 1920 m/s if you have correctly calculated Vt.

5. Compare Vt + Vesc (calculated in step 3) with the ÃŽâ€v calculated in step 4. This tells you how much you saved from Oberth effect.

[mhoram]

What is the proper way to calculate the ÃŽâ€v gained because of the Oberth Effect?

You want to calculate the gain of the oberth effect between two flight paths.

In your second calculation you compate the two flight paths

- Burn in LKO directly to a Jool-intercept

- Burn to get out of Kerbins SOI and burn to Jool after leaving Kerbins SOI

And the numbers match my experience.

In the description of your first calculation I can not see which two flight paths you do compare and also have problems understanding the reasoning for this equation:

ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2)

Perhaps the following helps you to see the connection to the escape velocity:

You can calculate the dv needed for a LKO-Burn to Jool via Escape velocities in the following way (which was also used in this thread to calculate ideal refueling orbits)

To do this calculation, the first step is to calculate the speed Ve you need at the SOI change.

For this use the Vis-Viva equation for an orbit around Kerbol with Periapsis @ Kerbin and Apoapsis @ Jool to calculate the speed at Periapsis.

Subtract from that number the velocity that Kerbin has around Kerbol and you get the velocity Ve =2760m/s.

Next step is to calculate the semi-major a=-468623 of the orbit of an object around Kerbin that has a velocity Ve at Kerbins SOI. (you can use the Vis-Viva euqation again)

And a third application of Vis-Viva with the semi-major a and Periapsis at LKO gives you the target velocity at Periapsis = 4233m/s.

Subtract from that number the current LKO-speed 2280m/s and you get in the 1950m/s range.

Edited September 26, 2015 by mhoram
typo

[Anglave]

1920 m/s is the burn you need to do to get from Kerbin to Jool, taking maximum advantage of Oberth effect. So if you want to know "how much benefit do I get from Oberth", you'd need to work out "how much more than that would I need if I didn't use Oberth effect". That's what Yasmy's doing, and I believe he's doing it correctly.

--snip--

I'm confused. Why would you do any calculation here? The answer is 1920 m/s.

This is the crux of my error, thank you. All this time I was assuming that the 1920 m/s value was ignoring benefit from Oberth Effect. I was trying to calculate the actual ÃŽâ€v one would need to expend (taking Oberth into consideration) to achieve the required 1920 m/s change, so as to find value of the savings.

Yasmy had the right idea all along. I was looking for a value lower than 1920, assuming some efficiency gain by considering the Oberth effect. Since the 1920 value already takes Oberth into consideration, I should instead have been seeking a higher number; the required ÃŽâ€v with "no Oberth", which Yasmy had already demonstrated how to calculate, here.

- - - Updated - - -

I'm not sure where you're getting the equation that I highlighted in red, above.

ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2)

From this page, about 80% of the way down the page, under the "Oberth Effect" heading. It was the most approachable math I found (including examples) that claimed to address the problem I thought I was trying to solve (how much ÃŽâ€v am I saving).

Edited July 9, 2015 by Anglave
added link to Yasmy's work

[Yasmy]

Seems like Anglave's equation, ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2) comes from http://www.projectrho.com/public_html/rocket/mission.php.

It is really not clear at all what the equation is for. It appears to describe a situation where you are already on a zero energy (KE = -PE) escape to infinity trajectory headed towards the sun at some far remove from periapsis, say at radius R. It then describes the velocity increase at radius R on the other side of periapsis after performing a burn at periapsis. It compares that to making an immediate burn at your current location. I'm not sure it's entirely correct. I get something slightly different. (Conserve energy before the burn as you go from R to periapsis. Burn. Conserve energy after the burn as you go from periapsis back to R. Find the periapsis burn needed to equal a local burn.)

A little further down on the same page is the more appropriate equation to use:

deltavee = √(v_inf^2 + v_esc^2) - v_cir

v_inf = velocity just before exiting Kerbin's SOI = 2.8 km/s

v_esc = Kerbin escape velocity = 3.2 km/s

v_cir = 2.2 km/s

deltavee = sqrt(2.8^2 + 3.2^2) - 2.2 = 2.0 km/s

The savings would be v_inf + (v_esc - v_cir) - deltavee = 2.8 + (3.2-2.2) - 2.0 = 1.8 km/s

This deltavee is close to the 1920 m/s figure we were using, and the saving are close to the 1.7 km/s I calculated before.

Note though that Snark is correct in pointing out that things are slightly different in the patched conic system KSP uses.

The equations from the linked article will be wrong where you cross SOI boundaries. They are fine for operating inside an SOI.

Edited July 8, 2015 by Yasmy

[mhoram]

ÃŽâ€v = sqrt(Vf^2 + Vesc^2) - sqrt(Vh^2 + Vesc^2)

I believe this equation is used in the case that (inaccuracies caused by KSP's patched conics)

- you enter Kerbins SOI with a speed of Vh

- want to exit its SOI with a speed of Vf

- want to calculate the dV needed to make this happen by a single burn at periapsis

So this formula does not fit to the situation you want to calculate.

[Anglave]

Thank you all for your informative and illuminating answers. I think I've a much better grasp of the situation now!

Humbling to realize I was just doing it wrong every which way, but uplifting to be helped so willingly by members of the community.