Problem Problem calculating Rama's orbit for use in Orbiter

[Peskie]

Greetings. This is my first time starting a new topic on this forum, I hope I put it in the right place. And I'll apologize up-front for the long post.

Recently I happened to cross paths with http://en.wikipedia.org/wiki/Rama_(spacecraft) and thought about adding a Rama vessel to Orbiter with a set of scenarios.

Rendezvous with Rama was always one of my favourite Sci-Fi books. Many years ago when first reading it I recall doing the calculations to check Clarke's description of the diameter, rotation, and centripetal acceleration and being pleasantly surprised that all the math worked out (i.e. Clarke had done the math when writing the book). So I figured I'd be able to use the books descriptions of Rama's orbit to create a matching orbit within Orbiter...

Has anyone else tried doing this? A forum and Google search turned up nothing.

I tried doing it and ran into difficulties. Basically the time from around Venus to the specified perihelion doesn't work for a hyperbolic orbit; the orbit would need to be elliptical for the time and distance specified (details below). Perhaps someone here can see if I've made a mistake in my procedure, assumptions, or understanding of the text.

Here are the relevant bits of information from the book that could be used to come up with an orbit:

• 31/439 Rama "was detected while it was still outside the orbit of Jupiter" (Chapter 2). That would be at >~5.77 AU from Sol. This was sometime in 2031.
• It's on a hyperbolic orbit relative to Sol; eccentricity > 1
• "Perhaps during the next few years some spaceship on its ordinary business might be routed close enough to get good photographs." (Ch2) This should give an approximate upper limit on the velocity/eccentricity (too high and it wouldn't stay in the solar system for years).
• "An actual rendezvous was most unlikely; [...] cutting across the orbits of the planets at more than a hundred thousand kilometers an hour." (Ch2) That's 27.78 km/s but it's not clear if that's out by Jupiter or by the inner planets (the wikipedia page seems to assume that it's the speed at detection).
• Several months later Rama is observed by a lunar telescope to have a rotational period of ~4 minutes.
• At some unspecified time later there was a meeting and then "Three months later, the space-probe, rechristened Sita, was launched from Phobos, the inner moon of Mars. The flight time was seven weeks [...]". "The two bodies would pass each other at two hundred thousand kilometers an hour." (Ch3) Once I've got the basic orbit figured out I could use this information to rotate the orbit around Sol to line it up for the specified intercept (e.g. tweak the argument of periapsis and possibly the inclination and LAN).
• Around a month later Endeavour rendezvous with Rama. It's unclear where Endeavour starts from, only that "the ship had been on a routine mission, checking and emplacing asteroid warning beacons" (Ch4); and "it had been necessary to rob three other ships [...] even with all the extra propellant [...] Rama was already inside the orbit of Venus when Endeavour caught up with it. No other ship could have done so." That's within ~0.72 AU of Sol.
• "In forty days they would reach perihelion, and pass within twenty-million kilometers of the Sun." (Ch4) That gives PeR = ~20e9 m = 0.134 AU and at around 0.72 AU from Sol a PeT = ~40 days = 3.456e6 seconds.

So I tried to come up with some orbital elements using the scenario editor to adjust the orbit of a random vessel to see what it gave.

• for simplicity I initially started with:
  • inclination of 0
  • longitude of ascending node 0
  • argument of perapsis of 0

these coould be adjusted later to have the path cross near enough to Mars for the specified intercepts.
​

• a perihelion of 20e9 m = 0.134 AU

Now I just need an eccentricity that gives:

1. Vel ~27.78 km/s at ~5.77 AU from Sol (if I use the value as the detection velocity as the wikipedia page does)
2. PeT ~3.456e6 seconds at ~0.72 AU from Sol

To get #1 needs an eccentricity of just a little under 1.07 (less if the specified velocity isn't reached until closer to Sol). This would then seem a good upper limit for me to use.

The problem is that to get #2 the eccentricity needs to be <1! According to the scenario editor, with an eccentricity of 1.0000001 (the closest value to 1 that it will let me enter) I get a PeT of only ~1.97e6 seconds (~22.8 days). (An eccentricity of 0.99 @ 0.816 AU still only gives a PeT of 2.14e6 seconds).

Is there *any* way to make a hyperbolic orbit that takes ~40d to go from ~0.72 AU to <0.134 AU from Sol? If I'm right and there isn't, does it seem reasonable to just go with an eccentricity of 1.05 so that in my scenario there would be only ~1.7e6 seconds=19.8 days from intercept to perihelion?

Given that this post has become quite long I'll leave the discussion on what to do with Endeavor's departure time and Rama's near Sol course correction to a future post.

Thanks!

 

[Mandella]

First off, this sounds like a really fun project that I will certainly download and enjoy! Now that I think of it, I'm surprised that no one has modeled this before, considering the influence Clarke has had on both this community and the real space program.

Contrary to wikipedia, I always took that speed figure to be a ball park estimate averaging the velocity of Rama throughout its course across the inner solar system. In any case, the forty day span from inside Venus' orbit to Solar close approach in rather important to the plot of the story, and therefore I think more important to preserve in the simulation, no matter what the speed finally comes to. Is it possible to keep the time and still have a hyperbolic orbit?

Now as to the Endeavor, I'm interested in what you come up with for its capabilities, as I always wondered if Clarke did not have to bend some rules of physics and engeenering to justify *any* craft being able to intercept and match Rama's course as described.

 

[Genius]

I remember that day I pick up the book. I could not put it down again!!! And I second that....it will be a awesome addonn; if done properly.
So all our wonderfull add on developers....here is a good challenge for you. Make it big...make it beautifull and let the Ramans come to our solar system.
In the same time it will be a good memorial for the late Arthur C.Clarke. He deserved that.

 

[Linguofreak]

Solar escape velocity at Venus' orbit is 50 km/s. If the orbit is hyperbolic, the ship needs to be going at least that fast when it crosses 0.72 AU. It will gain speed as it heads towards the sun. 50 km/s * 40 days is about 1.15 AU. What's more, solar escape velocity at 0.134 AU is 115 km/s. 115 km/s * 40 days is 2.65 AU. So the distance the ship travels between Venus orbit and perihelion needs to be somewhere between 1.15 and 2.65 AU, an educated guess says it's probably closer, but not too close, to 1.15 (say 1.5 AU). I doubt that there's any hyperbolic orbit that has a path length of even 1 AU travelling between 0.72 AU and 0.134 AU.

One question is this: does the book specifically mention that the craft is on a hyperbolic orbit? If not, it could have performed a breaking maneuver somewhere in the outer system and put itself on an elliptical trajectory. If so, Clarke screwed up.

Escape velocity is 27.78 km/s at around 2.3 AU, so since we're pretty sure that the orbit is elliptical, it has to reach this velocity somewhere inside that radius. Earth's orbital velocity is 29 km/s, so since the orbit extends well into the outer system we should assume that the v=27.78 km/s point is closer to 2.3 AU than it is to 1.

 

[Peskie]

First off, this sounds like a really fun project that I will certainly download and enjoy! Now that I think of it, I'm surprised that no one has modeled this before, considering the influence Clarke has had on both this community and the real space program.

And I second that....it will be a awesome addonn; if done properly.

I should clarify. My intention was to toss something crude together for myself (and anyone else that wanted it) that allowed me to fly the intercept and departure. I'm afraid that I have no experience with making meshes and or textures. I was going to try and make just a basic Rama shaped vessel, a long cylinder, and was going to omit the details on the "north" face and any textures. For the Endeavor I planing on 'borrowing' an existing craft and tweaking the parameters if/as required.

If however there is someone with experience making meshes and textures that would like to provide a Rama and Endeavor model I would of course use those.
My paperback copy of the book (a 1990 release I think) shows an artists version of the Endeavor and I could provide a scan if anyone is interested.

Contrary to wikipedia, I always took that speed figure to be a ball park estimate averaging the velocity of Rama throughout its course across the inner solar system.

From my initial post, if taken as the speed out by Jupiter it puts an upper limit on the eccentricity of a little under 1.07. So any eccentricity between 1 and that limit will most likely suffice and be reasonably true to the book.

In any case, the forty day span from inside Venus' orbit to Solar close approach in rather important to the plot of the story, and therefore I think more important to preserve in the simulation, no matter what the speed finally comes to.

I think for an Orbiter scenario the important thing is the distance from the sun at rendezvous and separation. Since orbiteers won't be spending weeks exploring the interior of Rama, if they have to separate in half the time I don't think it will be an issue.

Is it possible to keep the time and still have a hyperbolic orbit?

That was, in essence, my original question. I'm now fairly certain I was correct that there is no way to do this within the other constraints of the book.

Now as to the Endeavor, I'm interested in what you come up with for its capabilities, as I always wondered if Clarke did not have to bend some rules of physics and engeenering to justify *any* craft being able to intercept and match Rama's course as described.

I haven't gotten that far yet. Since there are no further details given in the book about Rama's orbit, or even the starting point for the Endeavor, I'll just make something up. I was planing on playing with the inclination, longitude of ascending node, and argument of perapsis to get Rama's orbit near enough to Mars for a reasonably efficient intercept. However, other than trial and error I'm not sure of the best way to go about this.

One question is this: does the book specifically mention that the craft is on a hyperbolic orbit? If not, it could have performed a breaking maneuver somewhere in the outer system and put itself on an elliptical trajectory. If so, Clarke screwed up.

He doesn't use the phrase "hyperbolic orbit", no. But the book makes it clear that the first time Rama does any maneuver that is observed is right at the end and that until then it is on a trajectory to leave the solar system.

At best the trajectory does a passive maneuver around Jupiter. I'd think that such a maneuver would have been to slow the vessel down (rather than speed it up) relative to sol to give Rama more time near sol. Since I'm working backwards on creating Rama's orbit, if I wanted to incorporate such a Jupiter pass I'd just need to adjust the orbit to pass close enough to Jupiter as if I was planing a Jupiter gravity assist to escape Sol.

This might give Rama a more reasonable arriving interstellar velocity. However, in the book when he talks about the scientists working Rama's path backwards to the nearest star system he doesn't mention any details about where Rama came from.

 

[Mandella]

Arg! I'm going to have to dig up my copy of the book when I get home -- I can't remember specifics (hey, I read it when it first came out!), but the wikipedia article says that the Endeavour caught up with Rama just inside Venus' orbit, and only stayed attached three weeks. Now I remember that it took them a few days to originally dock, and they detached some time before perihelion, but still 21 days is a lot closer to the math you've come up with.

One thing that also comes to mind is the possiblity that the book's "incorrect" math might come from typos that made it through the editing process. Not that it's *impossible* for Clarke to have just botched up the orbit, but I'd hate to think it of him!

 

[samplereturn]

Over the years, I've also tried "doing the math" and calculating specs to spacecraft or trajectories in Clarke's stories. As you correctly read, Rama's inbound orbit had an "e" greater than 1. Also, Rama made no course corrections from discovery until after Endeavor's separation.
Have you ever used JPL's Solar System Division site (//ssd.jpl.nasa.gov/) and their "Small-Body Database Search Engine" ? I found a practically parabolic comet with similar perihelia as Rama's ( <= 0.134 AU , from 20 mill/149.6 mill ) :


C/1948 V1 (Eclipse Comet) PeD= 0.135 AU, e= 0.999935, PeTime= 1948 Oct 27.43


When you plug this into their HORIZONS ephemeris generator, the comet travels the distance from Venus' orbit (~0.72 AU) to perihelion in approx 21 days, not the 40 days Clarke mentions. (The comet also passes Jupiter distance inbound, 5.20 au, in late Nov. 1947) If the trajectory was strongly hyperbolic, the comet would spend even less time in the inner solar system.

Wow, that's quite a difference... 21 days vs 40 in the novel! Clarke understood the mathematics of spaceflight quite well; how could he have missed this ? I guess you just have to start from farther out in the asteroid belt and rendezvous faster. After all, Endeavor had a mass of 5000 tonnes with propellant from three ships.