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Flag on Pluto - need help with math


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Hey everybody,

I want a flag on Pluto (using RO,RSS,TAC-LS,RT,kOS etc) and started by calculating different scenarios. It turned out, that Pluto is giving me some serious headaches. The biggest problem is the unneglectable inclination relative to the eclipse of 17° combined with Pluto`s excentricity. ---- I`m German, hope you understand what I write, feel free to correct me ---

 

Scenario 1 is just a Hohmann Transfer to meet Pluto when he crosses the ecliptic.

Scenario 2 is about meeting him at his Periapsis.

I do  not consider any swing-by maneuvers, because there would be to many odds...

 

So in order to calculate Scenario 1 I need to know my speed relative to Pluto, when I meet him at the Ascending Node so I can calculate the burn that keeps me in Pluto-orbit. But here`s the problem: how do I know Pluto`s distance from the sun and his speed when he crosses the Eclipse? Is there any "general" equation that would be usable for other inclinded planets, such as Mercury as well? 

 

Scenario 2 would require to burn my spacecraft`s apoapsis to Pluto`s Periapsis. His speed and my speed at that point are easy to calculate. The advantage of this scenario would be a much shorter flying time, and therefore less weight (less life support). What I save on Life support could be added as fuel and increase the Delta-V for the "Flip-burn". On the way back to Earth I wouldn`t have to do the flip burn and use the Delta-V to reduce speed just before Earth-reentry and keep g-forces lower, since I would not have to dive too deep into Earth`s atmosphere. But how do I calculate the "Flip-Burn"? Flip-Burn", because I would stay at the Eclipse until intersecting Pluto`s orbital plane and change the inclination ("flip" my trajectory) there. How do I calculate my speed and my distance from the sun when intersecting Pluto`s orbital plane with my AP at his PE ?

 

Any hints are welcome. Maybe someone has done and posted their math before, but so far I couldn`t find any.

 

Cheers, Whateverest

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For Pluto, you might want to send probes to test the scenarios. Probes should be launched first in order to see how much delta-v you need to get there, how much you need to slow down, and how much you need to land (if you want to land a probe first). Honestly, if you have sticky notes and probes, the math equations can be bypassed or at least a lot less needed. One last thing, you might want to at least TRY a gravity assist or two, because the transfer burn alone is almost 10 km/s.

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Going to Pluto with a Hohmann transfer orbit will take you 60 years and insane amounts of dV.

You best option is to use a gravity assist from Jupiter (it can help in matching the inclinations too): it will save dV and time. Gravity assists are fairly easy to plan using this tool.

For handwritten planning, you might want to look at the Vis-Viva equation (gives you your speed at a give altitude) or learn the maths of orbital mechanics (only way to deal with inclinations accurately AFAIK).

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If you can, I'd recommend waiting for a window to use consecutive gravity assists from Venus, Earth, Jupiter, and Saturn in that order. I have no idea how to calculate that precisely enough for your purposes but it is most likely an option for reducing delta-v.

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Hi! I'm no astrophysicist, but the challenge section is the place for trying difficult things, and this one sounds fun! I'll attempt to provide a way to calculate your first question: The distance from the sun and speed of pluto when crossing the ecliptic. As a note, I have a tendency to use odd words, so please (Please!) let me know if there's any points you'd like explained using different words. I've spent hundreds of hours looking up words in a dictionary for my second language and don't want to do the same to you. :)
 

Equations and Definitions
These are the 4 equations we'll need to solve your problem. I'll refer to them as Eq. 1, Eq. 2, and so on.
1. The argument of periapsis defines the angle between periapsis and ascending node.  https://en.wikipedia.org/wiki/Argument_of_periapsis
2. The ascending node and descending node are 180 degrees opposed.
3. The polar form of an elipse relative to one focus will provide the distance given an angle https://en.wikipedia.org/wiki/Ellipse#Polar_form_relative_to_focus
4. The Vis Viva equation (Thank you Leibnitz) is an energy balance to provide the velocity of an object given its radius. https://en.wikipedia.org/wiki/Vis-viva_equation

Symbols
a = semi-major axis  (Defines the size of an orbit, and is calculated by averaging the periapsis and apoapsis radii)
e = eccentricity (Defines the oval shape of an orbit)
Theta = angle of Pluto along its orbital plane relative to the periapsis. It's used in a cosine, so direction doesn't matter.
GM = The standard gravitational parameter, in this case referring to the gravitational pull of the sun. This is the product of two terms, and is typically referred to with the greek letter mu.

Values
We can find the important values for Pluto from Wiki: https://en.wikipedia.org/wiki/Pluto . I will use periapsis and perihelion interchangeably. I may also slip and say 'periapse' for some reason.... I have no idea where it came from.
Argument of Perihelion (Theta) = 113.834 deg
e = 0.24905  (unitless)

a = 5915 Gm (Gigameters, Notice the little 'm', It's not to be confused with GM, defined in a moment)
GM = 1.3271×1020 (has units. should be a force over a mass times an area)
 

Solving
Angle between periapsis and ascending node = 113.834                                     by definition (Eq.1)
Angle between periapsis and descending node = 293.834 = 113.834 + 180        by Eq.2

r(\theta)=\frac{a (1-e^2)}{1 \pm e\cos\theta}   (Eq. 3)

Plugging in to the above equation....
* Radius (distance) at ascending node = 5040.818 Gm = 5915 Gm * (1 - 0.24905 ^2) / (1 - 0.24905 * cos( 113.834 deg))   Here we have chosen the minus sign in the denominator because we're measuring relative to the periapsis. I'm assuming at any rate, because it gives the right numbers :D
* Radius (distance) at descending node = 6168.948 Gm = 5915 Gm * (1 - 0.24905 ^2) / (1 - 0.24905 * cos( 293.834 deg))   

v^2 = GM \left({ 2 \over r} - {1 \over a}\right)   (Eq. 4)

Taking the square root, and plugging in to the above equation...
* Velocity at ascending node =  5497 m/s =  sqrt( 1.3271 * 10^20 * (2/ (5040.8*10^9) - 1/(5915*10^9) ) )
* Velocity at descending node =  4537 m/s =  sqrt( 1.3271 * 10^20 * (2/ ( 6168.9 *10^9) - 1/(5915*10^9) ) )

Sanity Check
   Now we check our values relative to known quantities and see if they make sense! Hm. http://nssdc.gsfc.nasa.gov/planetary/factsheet/plutofact.html this should be an excelent resource. It says the orbital velocity varies between 3.7 and 6.1 km/s, so I'm willing to declare success. If anyone catches a mistake, please let us know. This isn't really my feild, and I'm doing the math quick before my little one wakes up. In any case, I hope this helps your trip to Pluto. Make sure to post if you make it there! And, thanks for the challenge!
 

Edited by Cunjo Carl
typo'd a number.
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....well, this left me speechless. I'm deeply impressed by the Support of all of you, especially the math done by @Cunjo Carl ! Thanks for spending your time on this. I'll try to understand the math in a silent minute, but the numbers "feel" right...

The suggestions of using gravity assists...well, I guess I won't make it without them. At least for the manned part of the Mission. Even those Kerbals don't live forever, do they? 

For the way back and for all the supply missions, I'll be using @Cunjo Carl's numbers. I'll let you know about my progress if you care. 

Right now I think of ways to survive Earth reentry at ~11000m/s and possible designs of my Pluto lander -the "JOHN WAYNE"- and where to put it. To save weight on the transitflight to Pluto JW will be waiting in Pluto orbit and pick up the Crew, bring it to surface and back up to the ship. Or a different ship...I'll let you know. Kids keep me busy  at the moment, so it could take a bit.

 

Thanks to all for your encouraging Support!!! 

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