Rendezvous with parabolic orbit

[BLANDCorporatio]

Ok, ultimately I'm interested in a rendezvous with an object moving on a parabolic orbit scenario, but baby steps. What options does one have to go from some circular orbit around the Earth to parabolic 'around' the Sun? As in, ways to actually plan to go from one orbit to the other, and compute deltav and time requirements of such plans?

(To make things 'simple', no gravity assists nor Oberth maneuvers except possibly with the Earth at the beginning. Ship needs to rely on its own power.)

All orbital maneuvers I see on wikipedia or Orbiter tutorials go between ellipses, which makes sense since those are actually useful. I need something a bit fancier though.

Can anyone help by pointing me to some good references and such? Thanks!

Cheers.

 

[statickid]

Not sure what you mean by "options."

Maybe this will help you to think about "transferring" from Earth to solar orbit.

1. Put a spacecraft in circular lunar-equatorial orbit.

2. Open two orbit MFD's.

3. Set one to MOON focus, and the other to EARTH focus.

4. Speed up the time so that your craft zips around the moon at a decent rate (100x)

5. Watch both orbits, you will see moon orbit looking round, while the earth orbit is constantly changing between highly eccentric, circular, and escape orbits.

6. If you are orbiting the Moon, you are also orbiting the Earth, however, the moon constantly changes your earth orbit.

7. Therefore, making an escape burn from the moon will only sometimes put you in an escape orbit from the earth.

The same idea applies to making an escape burn from the Earth.

When you are orbiting the Earth, you are already orbiting the sun. However, the Earth is constantly changing your solar orbit. One MAJOR difference is that if you ONLY escape Earth's orbit the direction you go will only slightly distort your solar orbit, because the Sun's influence is so great. In short, you have infinite options when transitioning from Earth orbit to solar-parabolic orbit. The details will be determined by the specific way you execute the maneuver.

---------- Post added at 01:44 PM ---------- Previous post was at 01:38 PM ----------

P.S. Orbits with eccentricity(e)

e=0: circle
0<e<1: ellipse
e=1: parabola
e>1: Hyperbola

So any equations you find dealing with eccentricity or orbit are not limited to only ellipses.

---------- Post added at 01:56 PM ---------- Previous post was at 01:44 PM ----------

p.s.s. I like to recommend flying around the Jovian or Saturnian systems before going on interplanetary flights. Try the Rescue on Io delta glider IV scenario if you have it.

 

[BLANDCorporatio]

Thank you for the time, but it's not quite what I'm looking for. Not interested in flying around in Orbiter just yet. Still at the phase where it's all meaningless buttons and panels ("open another orbit MDF" hth?), and it seems just needlessly complex when there are automatic Hochmann calculators out there.

That said-

1. Put a spacecraft in circular lunar-equatorial orbit.
{snip}
5. Watch both orbits, you will see moon orbit looking round, while the earth orbit is constantly changing between highly eccentric, circular, and escape orbits.

This doesn't quite make intuitive sense to me. If it orbits the Earth (or Moon) such that in an Earth (or Moon) centric system it closes a trajectory, there's no way it's going to zoom around the Sun (or Earth) in a parabola.

But anyway, I did try the DG in orbit around Earth, and switch ref to Sun. Lo and behold, the orbit looked predictably like an ellipse moving around the Sun.

You mention making an escape burn from the Earth (or Moon) might sometimes put one in an escape orbit around the Sun (or Earth). This also seems a bit weird. Earth's orbit is approximately circular (eccentricity 0.0167), which means most points are 'mostly the same' in what they'd allow you to do with things like escape burns.

Finally, I am familiar with parabolic orbits having eccentricity 1. However, as far as my limited knowledge goes, Hochmann and bielliptic transfers are meant to go from elliptical to elliptical orbit. They don't apply for what I want, and are too slow anyway.

Brachistochrone orbits are closer to what I'm looking for, and there's a way to compute deltav and other things about them at the excellent "So you wanna build a rocket" website, but it's again geared for planet to planet transfer. At some point in the calculation the (average?) orbital velocities of the start and target bodies are important.

It's getting late here, I'll come back tomorrow sometime with a worked example of a (semi-)brachistochrone and you can check my math.

Cheers.

 

[ADSWNJ]

Impressive first post, BLAND ! What's your goal with this maneuver?

 

[Cairan]

I can see a few uses, like catching to an Earth - Mars - Earth cycler, as proposed by Buzz Aldrin...

 

[BLANDCorporatio]

What's your goal with this maneuver?

I can see a few uses, like catching to an Earth - Mars - Earth cycler, as proposed by Buzz Aldrin...

Oh. That sounds like it might actually be useful. I'm intrigued by that idea, can you tell me more about it?

As for my goal, it's completely stupid. Basically I encountered a post from "Jules" on the RPGMP3 forums, and it inspired me to write a crossover Prometheus/Rendezvous with Rama ripoff.

*ducks*

Anyway, I wanted to get some numbers for possible maneuvers. I'm placing the action mid 2020s, so I'd like to use sorta-plausible (if speculative) nuclear propulsion rather than Thunderpants. Like, stuff we might get our hands on if we decided, we got ~10 years to get our act together and rendezvous with this once in a history object (before it decides to park itself in our doorstep anyways, if that's what it wants to do).

So I really need two maneuvers. One must be 'fast' (2-4 months) to catch up with the thing. Return to Earth can be slow because *handwave* they figured out suspended animation in the future.

I said I wanted to do some calcs. This really should have gone in a different post, but since I don't want to offend The Probe I'll continue here.

Ok, so I'll try to estimate deltav for a few brachistochrone transfers. I'll bold stuff I'm not sure of. I'll be using these equations from the "So you wanna build a rocket" website, Torchship section.

I'll assume my ship gets assembled in orbit around the Earth. I'll assume landing on the thing to intercept is negligible in terms of deltav, so the

totalDeltaV = transitDeltaV + matchOrbitDeltaV
transitDeltaV = 2*sqrt(distanceTraveled * acceleration)
matchOrbitDeltaV = abs( orbitalVelocity(target) - orbitalVelocity(Earth))
(orbitalVelocity here is orbital velocity around Sun)
acceleration = (4*distanceTraveled)/(transitTime^2)
(transitTime can be anything from 56 to 112 days)

For the purpose of these napkin calcs, let the target trajectory's semilatus rectum be 1AU, which results in (unless I forgot a factor somewhere) about 36 days to go from some point 1AU away from the Sun to perihelion, and 15 years from 34 AU (a bit beyond Neptune) to perihelion.

First thing: suppose I want my ship to match the target orbital velocity as the target intersects Earth orbit. Target's orbital velocity at that location is approx. sqrt(2)*orbitalVelocity(Earth).

Am I correct to assume then that
matchOrbitDeltaV = (sqrt(2)-1)*orbitalVelocity(Earth)?

If yes, it suggests there is a point on the target's orbit for which matchOrbitDeltaV = 0 (just choose a point 'higher' up). However choosing that point may require longer travel, so what I gain at matchOrbitDeltaV I lose at transitDeltaV.

Which brings me to the second point. How the h do I estimate distanceTraveled?

Suppose I launch (start the burn) on summer solstice. Should I then take distanceTraveled to be the Earth<->target distance at that moment? For example, suppose the target passes 'near' Earth at that day. Can I then say 'distanceTraveled' is 0, and I only need to match orbital velocity? (Which will require more than (sqrt(2)-1)*orbitalVelocity(Earth), of course, because I can't accelerate suddenly and by the time I catch up, the target moves faster)

Cheers.

 

[statickid]

You mention making an escape burn from the Earth (or Moon) might sometimes put one in an escape orbit around the Sun (or Earth). This also seems a bit weird. Earth's orbit is approximately circular (eccentricity 0.0167), which means most points are 'mostly the same' in what they'd allow you to do with things like escape burns.

A ship is in a circular orbit around a planet with a circular orbit. I forgot the arrows but in the diagram it would be orbiting anti-clockwise, and to be simple we'll just say in this star system it is prograde motion. The planet is the purple circle and the star is the orange glowy looking thing. The planets orbit is the blue line.

If you make escape burn 1 with whatever given Δv, your orbital velocity combined with the direction you escape will send you on a trajectory closer to the star (dark green orbit). Continuing to add MORE Δv, we'll say Δv+x, will allow you to fly closer and closer to the sun (pale green) but will not allow send you directly into a parabolic orbit (unless of course, you spend the absurd amount of Δv and actually keep burning until you start orbiting retrograde, and reach the stars escape velocity in retrograde direction.)

If you make escape burn 2 with exactly the same Δv as vector 1, It will send you on an orbit traveling farther away from the star, and if you continue to burn Δv+x, you will eventually be sent directly on a parabolic escape (pink line). If you want to enter a parabolic orbit with a periapsis very close to the sun in a simple way with somewhat efficient Δv, then you would escape from the planet using burn 1 and then apply vector 3 when you reach the periapsis of the pale green orbit yielding the orange trajectory.

 

[BLANDCorporatio]

That's more useful, thanks again. Any chance I can see the math behind the diagram, and get some deltav values? For example, when you mean first burn at vector 1, you mean just enough to escape Earth ((sqrt(2)-1)*orbitalVelocity(Earth), then for escape vector 3, just enough extra deltav to escape the Sun (whatever extra velocity is needed to equal parabolic velocity at perihelion)?

The transfer to orange that you describe is almost what I need, the problem is I can't have my ship get on the parabolic orbit at perihelion. That's a bit late. A couple weeks before the target reaches perihelion is the goal.

Cheers.

 

[statickid]

Hmmm, well the average orbital velocity of earth is 29.78 km/s according to the Earth wiki. I don't know what equation you're using but 41.42...% of earths orbital velocity would be approximately 12.34 km/s which is significantly higher than earths escape velocity of only 11.19 km/s as calculated by the equation Ve=√(2GM/r). I guess its even lower in orbit IIRC. So if you were orbiting in LEO at oh...say... 7.5 km/s, you would only need maybe 3.7-8ish km/s Δv (less?) to enter an escape trajectory and leave the Earth and get into the dark red or green orbits in the cartoon. The two hyperbolic trajectories would need quite a bit more (I called this number "Δv+x") to achieve a solar escape trajectory.

---------- Post added at 07:54 PM ---------- Previous post was at 07:37 PM ----------

If I may split hairs, your interest in brachistochrone torchrocket transfers seems a little bit at odds with your interest in plausible near tech from the 2020s don't you think ? I checked out the page you mentioned and I could be wrong but the equations there seem to be tailored for interplanetary flights with simple orbital velocities and easily calculable distances. I'd say the distance traveled is the distance between the location you left Earth and the location where you intercepted the other object.

 

[BLANDCorporatio]

Pure brachistochrones seem a bit unwieldy, yes. That's why I want to know what eqs to play with and tune a transfer that's reasonably fast, reasonably cheap.

Main issue I have, really, is getting on the parabolic trajectory a bit sooner than perihelion. {EDIT:} and really, that's the ultimate goal. If other curiosity questions from me are distracting/misleading, ignore them.{/EDIT}

Let's use your diagram for reference. Suppose I want my ship to eventually get on the orange trajectory. For your transfer, this happens when the ship reaches perihelion.

What I'd rather have is my ship settle on the parabolic orbit at such a point that, after following the parabolic orbit for two weeks, the ship reaches perihelion.

And I think you saw the page I referred to. Yep, those transfers are thought out for between planets. Everyone just does stuff between planets. Grr. Anyway.

A matter of curiosity. Suppose I have my ship, now in elliptical orbit around the Sun, 'far away' from everything else such that the Sun is the only important gravitational influence.

Which at Earth's orbit comes down to ~0.0057 m/s^2.

Suppose now I keep aiming my thrusters at the Sun and keep a sustained burn for a month. Intuition says (remember, keep thrusters aimed at the Sun and counterbalancing its gravitational pull) the ship travels in a straight line with constant velocity as net force on it is 0. What deltav do I expend on this outrageous maneuver? (Known: initial velocity, burn time, initial position from the Sun; this probably needs calculus to work out, but that's fine) {EDIT}(answer: time integral from burn start to burn end of thrust magnitude){/EDIT}

Obviously the hypothetical line is not an orbit transfer. I'd need to match whatever target orbit I want, resulting in some weirder thrust profile. But baby steps.

PS: the equation I used for the deltaV approx. was actually for solar escape velocity at Earth's orbit; got a little ahead of myself there

Cheers.

 

[statickid]

You could do something like this. Burn A puts you on the inbound elliptical orbit, and burn B would be a huge or long burn to match velocity. You would commence the earth escape path well in advance because you would be going much slower than the object until burn B. It would be timed so that y'alls ship intercepts the object at the moment your velocities match. I mean when you begin burn B, the object would still be pretty far behind, and from your perspective it would be falling at it reaaaaalllly fast and trying to slow down, tintin style.

One problem with this is that it is assuming the inbound object is fairly (exactly) matched with the ecliptic. In real life, extrasolar bodies don't have any good reason to be aligned with our ecliptic plane.

 

[BLANDCorporatio]

That's actually looking very good. So then, how much deltav does the hypothetical ship need for the maneuver? There's some deltav to go to the elliptical orbit around the Sun (so that's fine, I think I know how to estimate that, given some target green ellipsis), then there's deltav burn B. How do I calculate that?

One problem with this is that it is assuming the inbound object is fairly (exactly) matched with the ecliptic. In real life, extrasolar bodies don't have any good reason to be aligned with our ecliptic plane.

That's ok. That the target is 'close' to the ecliptic (and 'close' to parabolic in trajectory) is one thing that has in-story-everyone suspicious. The target otherwise looks like a dead ship, but as you say there's no reason for a random outer space roaming object to be closely aligned to the ecliptic.

Cheers!

 

[statickid]

:hmm: well wouldn't it just be the difference in your velocities at that point? Plus extra for any plane changes. You'll have to settle on the specific geometry you want to work with and where you want them to intersect. Maybe you can write a supplementary book detailing the interpersonal relationships of the mission planning astrophysicist team.

 

[BLANDCorporatio]

Ultimately I think I'll settle for seeing what impulsive accelerations I need to apply at the two points, and get a quick estimate for minimum deltav thusly. Of course, real engines can't apply Dirac impulses of thrust. So I'll then need some kind of fudge factor applied to the previously estimated deltav.

Lol at the last suggestion, but alas no.

What I do want is just to have a somewhat workable/plausible background detail- how long did the trip take, what kind of engines were needed. I could guess that the answers would be "a long while, just to catch up with something billions of miles behind" and "the most furious nuclear firecracker ever", but a bit more guesstimating can't hurt.

Cheers.

 

[dgatsoulis]

I did a few calculations to see what kind of dV it would take to rendezvous with such a target.
Given:
A) The target's semi latus rectum = 1AU, meaning a perihelion of 0.5 AU
B) Target's eccentricity = 1
C) The target is completely coplanar to the ecliptic.
D) Parking orbit of spacecraft around Earth = 200x200 km altitude.
E) Freedom to select any angle between the target and Earth at the beginning of the transfer.

I did the first by hand, looking for a minimum dV, 2 burn solution, with the encounter exactly at perihelion. The result was ~15.5 km/s total dV, with a Time Of Flight = 118.62 days.

But the OP mentioned an encounter 2 weeks before perihelion, so I fired up IMFD and started looking for a minimum dV solution. The best I could find was this:

With Earth excess V = 5936 m/s, meaning an Injection dV of 4735 m/s from a 200km orbit and 12760 m/s for the second burn, gives a total of ~17.5 km/s .
The TOF was 90 days, with the encounter exactly two weeks before the target reaches perihelion.


Just for the fun of it, I then tried to find the minimum TOF solution. The result was much better than I expected. With the launch when the target and the Earth are at about the same position, the result was an Earth excess velocity of 29.5 km/s, meaning a dV 23.7 km/s and a soft encounter that only required ~200 m/s. This came to a total of 23.9 km/s for a TOF = 25 days. (The encounter was still set to be 2 weeks before perihelion).

Not very realistic, because if the target passes close from Earth it will affect its trajectory, but fun to setup and try to find a good solution.

I hope these numbers help.

 

[BLANDCorporatio]

Well now, IMFD sounds like a great add-on. I'll join the dark side, I mean Orbiter, then

Thanks a bunch for those examples. I'll avoid sending the target too close to Earth, precisely because I don't want to have to handle trajectory changes too. So with some rejiggering I'll get my numbers.

But anyway, they'll probably not be far off from those, and somewhere between 17.5km/s to 20km/s for the trip in one direction- not as bad as I feared, especially if *handwave* (fusion) nuclear engines with exhaust velocities about equal to the total deltaV to go and return. 90 days transfer? Quite ok, especially if *handwave* suspended animation.

Cheers!