A handy chart for Aerocapture at Jool

[alterbaron]

Hi all,

Here's a little something I made for fun this afternoon: it's a Jool aerocapture chart.

Big Version

This chart gives the periapsis distance for aerocapture into a variety of different orbits after entering Jool's SOI.

To use it, exit time acceleration once in Jool's SOI. Find the curve corresponding to the orbit you want after aerocapture. Find where that curve intersects your orbital velocity (measured on the navball). The y-axis reading then gives the periapsis distance that should aerocapture you into that orbit.

This chart should be enough to get you reasonably close (delta-v wise) to any target orbit. For calculating other aerobraking maneuvers, I'd suggest using my aerobraking calculator: http://alterbaron.github.io/ksp_aerocalc (The plot was obtained using the same approach used there.)

(NB: This chart assumes that your ship does not have lift surfaces. These might mess up your results if you have any. Use right at SOI entry for best results.)

Let me know what you think!

Edited May 22, 2013 by alterbaron

[CaptinKornflake]

Oh my god man this is one of the most amazing things I have ever seen!

THIS IS SOME ACTUAL SCIENCE TO DO WITH KERBAL SPACE PROGRAM!

Because remember kids... the difference between science and screwing around is writing it down.

[patrickcanttype]

Not just science; color coded science! With a graph!

This looks very useful, and I'll have to save that calculator to my favorites!

[Specialist290]

Ooh, nicely done. I'll have to knick this for the Drawing Board

Incidentally, I've been trying to crack the math for aerobraking myself, but I've had a bit of trouble lately. Unfortunately, calculus is one of the subjects I've never studied, but I'm pretty good with college-level algebra, and I can do some stuff with geometry (including conic sections). Think you could help me demystify some of it?

[alterbaron]

Incidentally, I've been trying to crack the math for aerobraking myself, but I've had a bit of trouble lately. Unfortunately, calculus is one of the subjects I've never studied, but I'm pretty good with college-level algebra, and I can do some stuff with geometry (including conic sections).

You don't need to know much calculus to do these sort of calculations numerically (which is what I did here.) But you do need some comfort with forces in physics.

Basically, draw the free body diagram for the craft. You'll have a force due to gravity, and a force due to air drag. You take the net force from the free body diagram and set F=ma, or a = F/m.

Once you have the acceleration as a function of position and velocity, you can "plug the formula into a computer program to have it numerically solve for the path that the object will take."

I say this loosely, because the book-keeping can get rather tedious. But it's simple at heart.

This number crunching is based on calculus and can be summarized as "step forward in time, re-calculate forces, update position and velocity, and repeat."

Everything else is just a bit of tricky geometry.

[NeoMorph]

I cheat... I use the predictor in MechJeb's landing system.

[alterbaron]

I cheat... I use the predictor in MechJeb's landing system.

That's true, but wouldn't you feel more like a rocket scientist using a chart?

[Pyotor Gagarin]

This looks incredibly useful. I love that the SOI-transition velocity scale goes up to 5000m/s!

[alterbaron]

This looks incredibly useful. I love that the SOI-transition velocity scale goes up to 5000m/s!

Well, you never know what people will do . . .

[alterbaron]

Here's some mathematical details, for those who are curious:

Let r = (rx,ry) be the vector from the center of the planet to our ship, and v = (vx,vy) be the velocity vector of the ship.

Then the force of drag is given by the following equation:

Kp = 1.2230948554874*0.008

FD = -0.5 Kp P0 exp((R0-norm®)/H0)norm(v) d m A v

Where Kp is the conversion factor between pressure and density given on the wiki, P0 is the pressure at sea level (1atm for Kerbin), H0 is the atmosphere scale factor (5000m for Kerbin), R0 is the radius of the planet (needed since the vector r extends from the center of the planet), d is the coefficient of drag, m is the ship's mass, and A is the "cross-sectional area" of the ship, which is a constant in-game. The velocity v is a vector, don't forget!

Note that in the real world, there is no factor of "m" in the above equation.

The force of gravity on the ship is simply:

FG = - m(mu/norm®^3)r

Where mu is the gravitational parameter of the planet (3.53 x 10^12 for Kerbin).

Note that r is a vector!

Then the total force on the ship is just F = FD + FG. (A vector sum).

Note that norm® means the vector (cartesian) length of r.

These equations are used to simulate the trajectory of a ship as it passes through a planet's atmosphere. Here's a quick method that works o.k.:

Say the ship has a current position r=(rx,ry) and a velocity (vx,vy). We calculate a net force of F=(Fx,Fy) using the above equations at the current position and velocity.

Our acceleration is, by newton's law, a=F/m

Call our current time t. Consider a small time step dt. The velocity at time t+dt is approx. v+a*dt.

Likewise, the position is approx. r+v*dt.

Using this new approximate position and velocity, we can calculate F anew. Do this, rinse, repeat, and you're pretty much done. If you hold on to all your old values of "r" as you go, you have the path that the ship followed through the atmosphere. You also have its velocity at every point in time.

Neat, huh?

Edited May 22, 2013 by alterbaron

[Akalaël]

Nice one for sure.

[TMS]

Excellent.

Really like the website calculator.

Would make it even more intuitive if it had the option of using pre-defined target apoapses (so selecting Jool as your current orbital body gave the apoapses of it's moons), along with the existing customised apoapses.

Thanks for this.

[bsalis]

So you can use this to set your Ap above Jool to match the altitude of any moon. How do you know it will be there when you are to intercept the SOI?

[TMS]

^ You wouldn't, but when reaching your desired post-aerobrake elliptal apoapsis, you'd raise your periapsis out of Jool's atmosphere and orbit until you intercepted (and hope you don't have an encounter with any inner moons).

[Daze]

Amazing work! Very useful, thanks

[Kosmic Debris]

This answers a question I had in another thread - awesome!

Thanks!

[SkyRender]

A good start! Now we should develop some similar aerocapture charts for Kerbin, Duna, and Eve. Fortunately, none of those three have as many targets to consider. But it can still be useful to know, for example, that you can get into a fairly low Kerbin orbit coming back from the Mun by having an aerocapture periapsis of 34.7 to 35.1KM.

[Titus Jeb Worshipper]

Because remember kids... the difference between science and screwing around is writing it down.

This made me chuckle I never get tired of seeing the proper science done on these forums, I understand none of it, but its nice to know that someone, somewhere knows what is going on

[Specialist290]

SkyRender: The Drawing Board has a link to the thread where I posted a chart with the altitudes for various atmospheric pressures. That might come in handy, as will the scale height equations on the wiki I used to create it.

[AlexisBV]

Here's some mathematical details, for those who are curious:

FD = -0.5 Kp P0 exp((R0-norm®)/H0)norm(v) d m A v

Where Kp is the conversion factor between pressure and density given on the wiki, P0 is the pressure at sea level (1atm for Kerbin), H0 is the atmosphere scale factor (5000m for Kerbin), R0 is the radius of the planet (needed since the vector r extends from the center of the planet), d is the coefficient of drag, m is the ship's mass, and A is the "cross-sectional area" of the ship, which is a constant in-game. The velocity v is a vector, don't forget!

I don't see the planetary rotation in the equation - you're assuming that the atmosphere is static around the planet, and not rotating with the surface. This could potentially make a dramatic difference in drag between a polar, prograde or retrograde aerobraking. In the current equation they'd all be the same. Has this been verified?

I'm pretty sure the atmosphere rotates at the same angular velocity as the surface (at least at surface level), otherwise we'd have winds of 174m/s at the KSC and everything would be blown off the pad...

[tntristan12]

I've seen a chart like this before, and I've been looking everywhere for it ever since! Thanks man