Why New Horizons Won't Orbit Pluto (with math and shit)

"Racescort666" (Racescort666) 07/16/2015 at 10:29 • Filed to: New Horizons, Pluto, Math

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You may or may not have seen the !!!error: Indecipherable SUB-paragraph formatting!!! but I was hoping that there would be some math in this article. Undeterred, I have decided to do it myself. Below is a back of the napkin mathematical explanation of why New Horizons won’t orbit.

First, let’s start with the stuff we already know about the mission. We know the current velocity of the probe, the mass of the probe, the fly-by distance, the radius of Pluto, and the mass of Pluto. These values can be used to define a hypothetical orbit mission:

NH Vel: 14,000 m/s (from article)

NH Mass: 478 kg ( !!!error: Indecipherable SUB-paragraph formatting!!! )

NH Dist: 7,800 km (wiki)

Pluto Rad: 1185 km (wiki)

Pluto Mass: 1.305×10 22 kg (wiki)

We know that 14 km/s is greater than escape velocity for Pluto since the current mission is a fly by. Think of it like rolling a ball in a funnel, if the ball is going too fast, it shoots out the other side. That’s exactly what New Horizons is doing now. What we need to do, is figure out how much it needs to slow down in order to not go flying out but not too much so that gravity takes over and it just crashes into the surface.

If New Horizons were to orbit at the altitude of it’s fly-by, it would need to slow down to the speed that corresponds to that orbital altitude. That velocity is determined by the following equation:

G is the gravitational constant: 6.674×10 11 Nm 2 /kg 2

M is the mass of the parent body, Pluto which we listed above and R is the orbital distance from the center of mass. For simplicity’s sake, I’m going to assume Pluto is a perfect sphere and the center of mass is exactly at the center. So that’s the Plutonian radius 1185 km plus the fly-by altitude 7800 km, 8985 km.

Chugging the numbers, the approximate velocity for that orbit is 311 m/s. This is way slower than the fly-by velocity. For reference, the ISS orbits the Earth at an altitude of around 410 km and has a velocity of about 7660 m/s.

So now we need to figure out how to slow New Horizons down to achieve orbit. Enter, the rocket equation. This equation helps you determine a change in velocity with a corresponding change in mass for a given exhaust velocity (or engine efficiency as the case may be).

This is best expressed in terms of specific impulse and solving for initial mass.

ve can be represented as Isp * g0 (exhaust velocity = specific impulse of the engine times standard gravity).

I couldn’t find an image of the equation the way I wanted so here goes with regular characters:

m0 = m1 * e^(dv/(Isp * g0))

If we’re extremely generous and use New Horizons’ current mass as the final mass (m1) and also somewhat optimistic and say that New Horizons has an engine with a specific impulse of 400s, we can calculate what the initial mass would be. For reference, the Space Shuttle Main Engines (which are pretty efficient rocket engines) have a vacuum specific impulse of about 450s. A thruster for this type of mission would have a much lower efficiency.

So, dv = 13689 m/s, m1 = 478 kg, Isp = 400 s, and we get roughly 15647 kg for m0.

New Horizons would need over 15,000 kg of propellant to slow down to an orbital velocity around Pluto and that’s without trying to figure out how much structure would be needed to support that much propellant. That’s the size difference between a piano (which New Horizons has already been compared to) and a small semi-truck.

I actually posted this whole thing !!!error: Indecipherable SUB-paragraph formatting!!! over on the Giz article but since I spent some time looking up values and making calculations, why not share it with everyone on Oppo too.

djmt1 > Racescort666 07/16/2015 at 10:32

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I bet you’re good at Kerbal Space Program.

Racescort666 > djmt1 07/16/2015 at 10:35

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I’m not terrible but there are plenty of people who are much better than me. I’m kinda lazy when it comes to KSP and use the calculators for delta-v and orbit transfers. Although, knowing your delta-v budget helps a lot for building spacecraft.

ly2v8-Brian > Racescort666 07/16/2015 at 10:36

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Way to math the shit out of that.

Racescort666 > ly2v8-Brian 07/16/2015 at 10:37

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I do what I can.

Stapleface-Now Hyphenated! > Racescort666 07/16/2015 at 10:38

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I kinda like math, but this shit hurts my brain.

Still though, I was kind of wondering why only a flyby mission. Now I know.

The SciFi nerd in me wonders if it's going to encounter some evil space warzone as it journeys further into the Kuiper Belt, kind of like the Reavers on the way to Miranda.

ly2v8-Brian > Racescort666 07/16/2015 at 10:45

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That said I’m pretty sure NH’s mission wasn't to orbit Pluto but to make continuous passes by it and other Kuiper Belt Objects. So I'm not sure where the talk of setting it in orbit came from.

DoYouEvenShift > Racescort666 07/16/2015 at 10:50

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I thought that article was pretty silly too. Thanks for taking the time post it. I wish so bad I could sit down and play Kerbal. But with working damn near 60 hours a week. No time haha.

Racescort666 > DoYouEvenShift 07/16/2015 at 10:54

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I hear ya. I’ve been there, working crazy hours and getting home going right to bed. Things are somewhat slow now and I’ve been playing with spreadsheets for a while. Doing the math for this was a little bit more of the same.

unclevanos (Ovaltine Jenkins) > Racescort666 07/16/2015 at 11:03

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I eat differential equations for breakfast! We need more math in oppo.

Racescort666 > unclevanos (Ovaltine Jenkins) 07/16/2015 at 11:09

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I am happy to provide.

DoYouEvenShift > Racescort666 07/16/2015 at 11:40

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I think the math is fun too haha. I sucked at math in school, but when I can apply it to something I like its a lot easier. I live really close to KSC so I try to make it out to as many launches as I can. This coming Wednesday is night launch of a Delta IV. Cant wait to see that one.

MRtwo > Racescort666 07/16/2015 at 11:46

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nvm

Grindintosecond > Racescort666 07/16/2015 at 11:48

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Too much MATH!!!

uofime-2 > Racescort666 07/16/2015 at 11:57

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so basically the problem is that it is small and far away we can use the slingshot trick to gain velocity to get there in reasonable time, but then we are going way to fast once we get there and it is way to small to use a similar technique to slow down.

Santiago of Escuderia Boricua > Racescort666 07/16/2015 at 11:59

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Yeah it’s not like we couldn’t make a probe orbit Pluto, but it has to be going pretty slow so there’s no way we could do it in just 9 years

Racescort666 > uofime-2 07/16/2015 at 12:27

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Pretty much. The best way would be to use a rocket of some kind to slow down. You could probably aerobrake a little too and help cut down on how much fuel you’d have to bring with you. By and large, you wouldn’t be able to gravity assist into an orbit in a reasonable timeline.

Racescort666 > Santiago of Escuderia Boricua 07/16/2015 at 12:29

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I ran some quick numbers with the idea of using an ion engine like Dawn and the propellant mass is much more reasonable (500-600kg). Although I have no idea how long it would take to slow down 13.7 km/s

uofime-2 > Racescort666 07/16/2015 at 12:32

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and as you showed the size/cost of the rocket to do it would be obscene.

Which brings us to the problem with Newtonian based space travel. Damnit!

Santiago of Escuderia Boricua > Racescort666 07/16/2015 at 12:44

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I mean that if it was the intention to orbit, we wouldn’t have done an approach at that speed to begin with. It’s hauling ass right now because we wanted to get to Pluto ASAP

Patrick Nichols > unclevanos (Ovaltine Jenkins) 07/17/2015 at 10:36

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sigma squared is always open (though its currently just me) and I have been inconsistent with my posts...

I should probably get the word out a little more, might wait for next weeks cross thread

unclevanos (Ovaltine Jenkins) > Patrick Nichols 07/17/2015 at 11:46

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Oppo has some engineering and stem people here (mostly engineers). It shouldn’t hurt to cross post more often unless they’ve been scarred by the hell they call Calc 2.

ttyymmnn > Racescort666 07/29/2015 at 16:54

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Is this going to be on the test?

Racescort666 > ttyymmnn 07/29/2015 at 16:57

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The test is open book so all you need to know is that the formulae exist and what they’re used for.

OPPOsaurus WRX > Racescort666 08/28/2015 at 10:30

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Where is it now?

TipsyT-Rex > ly2v8-Brian 08/28/2015 at 10:30

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Yeah, that’s why he did this, to demonstrate exactly how impossible it would have been to stop new horizon and drop it in orbit.

StingrayJake > Racescort666 08/28/2015 at 10:36

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(I know this is an older post but it’s in the popular posts today...)

Short answer: It would take a helluva lot of energy and fuel to slow down enough to go into orbit.

Racescort666 > StingrayJake 08/28/2015 at 10:44

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No big deal on posting to an older post. I commented with a link to it on the FP today so I was ready for new comments.

Racescort666 > OPPOsaurus WRX 08/28/2015 at 10:46

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On its way out of the solar system. Or rather, on its way to other Kuiper Belt Objects, which haven't been determined/announced as far as I know.

Sean Melchionda > Racescort666 08/28/2015 at 10:46

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So essentially we would need another booster which has been kept in reserve for the exclusive purpose of slowing this thing down. 15,000 kg really isn’t that much, but to get that extra booster out to Pluto would require a very different kind of mission. For example maybe we would launch several booster stages into orbit, have them join together in orbit and then those boosters would push new horizons and its “braking booster” off to Jupiter. It would be much more expensive than what we did... But we would be left with a satellite in the Plutonian system.

OPPOsaurus WRX > Racescort666 08/28/2015 at 10:50

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it kinda sux, it passes Pluto and we dont hear anything about it any more. I thought they were downloading info for the next 6 months or something.

Racescort666 > Sean Melchionda 08/28/2015 at 10:53

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Compared to the size of the spacecraft, 15,000kg is a ton of propellant. (15 tons in fact, huehuehue). I played around with the numbers a bit and it would need a much more efficient engine than a chemical rocket to be even remotely practical.

Racescort666 > OPPOsaurus WRX 08/28/2015 at 11:02

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Yeah, so NH doesn’t have the bandwidth to transmit all of the data it has received at once (plus the 4.5 hour transmission time doesn’t help). So it’s going to take a long time to download all of the data.

The initial transmission was considered high priority which is why we have the images we do. For the next year or so, they will be transmitting all of the data uncompressed (as opposed to compressed like the current data) with the additional issue of the deep space network can only receive data at certain times of day (that is, when the antenna is pointing at Pluto). As the distance increases, the bandwidth drops too so this is a factor in the time. I think I heard somewhere that they have something like several gigs of data that they have to transmit at speeds that make 14.4k modems look like fiber optic.

Sam > Racescort666 08/28/2015 at 11:19

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Can you math something else?

How long would it have take New Horizons to get to Pluto, if they had sent it with the intention of slowing for an orbit?

MasterMario - Keeper of the V8s > Racescort666 08/28/2015 at 11:41

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yes, orbital mechanics! honestly it is actually pretty simple math, it was one of my easier engineering courses. now satellite mechanics...if you ever want to confuse someone show them some of the multi-page equation derivations from that course

Brianorca > Racescort666 08/28/2015 at 11:42

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When we launched it, we had no idea what Pluto’s atmosphere looked like, so an aerobrake would be out of the question. (Now that we have seen it, it’s probably way too thin to be useful, even though it’s 10x denser than we first thought.)

Brianorca > Racescort666 08/28/2015 at 11:53

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An ion engine would also require a much larger power source. Dawn could use solar panels because it stays relatively close to the sun. New Horizons with an ion engine would need a much heavier nuclear power source. It would probably need more nuclear fuel than exists in the current NASA stockpile. (If you stayed with the RTG design.)

Racescort666 > Brianorca 08/28/2015 at 12:10

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Future mission.

Racescort666 > Sam 08/28/2015 at 12:23

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I could look into it. That's not something I know how to do off hand though. Essentially, you'd want a regular Hohman transfer type orbit to save on fuel which could take as long as 100 years. New Horizons was envisioned with low transit time in mind which is why they have such a crazy fly-by velocity.

d15b > Stapleface-Now Hyphenated! 08/28/2015 at 16:24

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Star for SciFi-ing it out.

NotUnlessRoundIsFunny > Racescort666 08/28/2015 at 21:11

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You're like, smart and shit.

The Artist Formerly Known As... > Racescort666 08/28/2015 at 21:46

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I’m lousy at math. But this rocks, and it was pretty clear to me. Very cool.

Racescort666 > The Artist Formerly Known As... 08/30/2015 at 08:43

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Thanks!

nwss > Racescort666 11/25/2015 at 09:53

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e is amazing isn’t it? It’s more prevalently used than pi for sure.

Before I read this article, I was like “I bet e is going to be used somewhere, I was right” haha