Trying to derive some physics equations

Factual Questions
1

I was writing a story that involved a space ship getting stranded in space and being pulled towards Jupiter by its gravity. I wanted it to be as realistic as I could make it, so I tried to do the math and figure out the time it would take to reach the atmosphere and the speed when it did given an initial speed and distance. I realized I didn’t have enough math knowledge. So I bought a book and learned calculus. That didn’t help, since gravity is related to distance and not time. I really wanted to figure this out myself but it seems I’m just not capable. So I turn to you guys. What I need is an equation where I can figure out the speed and amount of time it would take a ship to to reach a planet given an initial distance and velocity.

Also, the ship falls partly into the atmosphere. It would help to also have an equation that calculates what the pressure is at a specific altitude. This one wouldn’t have to be very precise. It could assume a constant gravitation force and atmospheric composition. I know the pressure will be equal to the weight of the gasses directly above the object, and density will be proportional to pressure, but coming up with an exact equation is beyond me.

So I give up. I could design a computer simulation for the first problem that estimates it by assuming constant acceleration through each of many divided sections along the way, but that wouldn’t be easy or ideal. How about a little help from the physicists out there?

2

Not sure about the falling into Jupiter thing - I never studied orbital mechanics. But the pressure one is not too tough. The simple formula for pressure is
P=ρgh where ρ (rho) is density of the fluid, g is the acceleration due to gravity and h is the height of the fluid column.

In actual practice, both ρ and g will change with altitude, but can be assumed constant for relatively small changes of h.

As for the density of the Jovian atmosphere at a given altitude - I haven’t a clue.

3

Some data on the Jovian atmosphere here.

4

First of all, while I appreciate the extent you’re willing to go to for technical accuracy, you’re making this way too complicated. It’s all well and good for Heinlein to boast how he and his aerospace engineer wife spent three days and yards of paper calculating an orbital ascent in order to get one line of dialogue, but unless you do something utterly foolish–like have the Earth rotating the wrong direction[sup]1[/sup]–nobody without an advanced degree in aerospace engineering or astrophysics is really going to take notice (and in fact, few that fit that description will). Just SWAG[sup]2[/sup] it and go on with life.

That being said, given a distance and initial (vector) velocity, it’s a simple application of Kepler’s Laws to figure out the free unpowered orbit of a spacecraft. I say “simple” in terms of someone who regulary plays with orbital mechanics or toys around with Matlab/Scilab as a hobby[sup]3[/sup]. Even if you don’t, the calculations aren’t that hard, though you’re going to have to take into account tidal effects, the effects of the Galilean moons (we can dispense with the smaller outer moons out of hand), and of course, as you note, atmospheric effects in computing the resultant degenerate orbit. There’s no easy closed form solution to that; any generalized n-body solution, and especially one with variable forces, is going to a) have a large family of solutions, and b) be highly perturbative (i.e. very sensitive to initial and boundary conditions) and so will require interative simulation. If you want a reference, I think highly of Prussing’s Oribtal Mechanics, but seriously, anything you need to know for this calcuation you should be able to get from a basic university-level physics text.

However, if you fail to give complete detail on velocities and times, nobody is going to be able to calculate squat, and unless you make some egregious error, even the most pedantic physics grad student will give you a pass so long as the story is interesting. And since your characters are (presumably) going to suffer a hideous death in the crushing pressures of Jupiter’s inner atmosphere (the pressure-distance relationship isn’t linear, by the way; beyond the Jovian upper atmosphere very little is known about its actual composition, so feel free to make up things as you go along) then nobody’s really going to be checking up on them anyway. Rhubarb’s calculation is for an incompressable fluid (derived from Bernoulli’s Principle) but if you need a number you might as well use this as anything else. It won’t necessarily be any more accurate than pulling a number out of the air, but at least it’ll have the benefit of having a totally indefensible calcuation behind it, which is sufficient for NASA Safety Management[sup]4[/sup].

Just write your story. Run a few rule-of-thumb Kepler equations if it’ll make you more comfortable (or offer up some specifics here and you’ll have a dozen engineering/physics geeks running Matlab simulations of your scenerio thirty minutes after you hit submit) and get on with the business of describing your characters.

Stranger

[sup]1[/sup]As in the first edition of Ringworld by Larry Niven. Never mind that he totally missed the point that the Ringworld is unstable.
[sup]2[/sup]Scientific Wild Ass Guess. It’s a technical term. Trust me.
[sup]3[/sup]Guilty.
[sup]4[/sup]Seriously, don’t ask. It’s been a hard week

5

First, I agree with Strangers’ assessment. There’s been a lot of good SF by people who don’t know any science (Fredric Brown, for one. Jack Chalker for another), and you can get away without having to calculate things beyond rough numbers to avoid howlers (100 years to get to Jupiter? Five Minutes to the sun?)

But this sentence in your OP bugged me:

It sorta suggests that calculus won’t help you with distance problems, which isn’t true – although that may not be clear right now. Calculus is the mathematics of changing rates, and it works if the rate is taken with respect to time or to distance. In fact, the time and distance of your journey are intimately tied together.
Therre are a lot of books that will help you learn orbital mechanics, but I recommend getting hold of The Feynmann Lectures on Physics. In the first volume he works his way through plotting an orbit using just the basics, without getting lost in the equations, and it shows you what’s really going on. I once judged a science fair in which one kid worked out the dynamics of something traveling to/from Jupiter using Feynman’s method and plotting it up on a computer.

6

To expand upon that, you’d need to integrate over the acceleration due to gravity as a function of distance with respect to time (since the acceleration will change with distance, and distance with time) to get velocity, and accelerate over velocity to get position. But that’s messy; Kepler will get you conveniently in the ballpark, if you’re willing to ignore the (likely minor) effects of the Galilean moons. And once you hit atmo its all going to come apart like a cheap gold watch anyway.

Stranger

7

Don’t worry about the Galilean moons unless it’s convenient for your story. The Keplerian calculation will give you a time to hit the upper atmosphere, given particular starting conditions. For any given set of starting conditions, there’s guaranteed to be some arrangement of the moons such that the time until impact will still be the same as the Keplerian solution. And unless you give way, way too much detail, nobody will ever be able to say that the moons weren’t in such a configuration at the time.

About the only significance the moons could have for your story would be that interaction with one might turn an apparently-stable orbit into one which plunges into Jupiter, which could be how your characters got into their predicament in the first place. But a one-off remark about the rookie navigator forgetting to take into account Ganemede’s gravity would be plenty, there: You don’t need (and probably don’t want) to give a complete ephemeris explaining the situation.

8

I should, but I’d really like to have an estimate at least. I’ll ignore the gravities of the moons since they won’t be very close anyway, but I’d at least like to be able to calculate how much Jupiter’s gravity would accelerate something by the time it reached the atmosphere. Plus when something mathematical seems just outside of my reach, it bothers me and I tend to spend a lot of time trying to figure it out.

I may have overlooked something, but I don’t see any equations there that help me. They all seem to assume that the major-axis is known, but it isn’t since the ship is initially travelling toward the planet.

[quoteAnd since your characters are (presumably) going to suffer a hideous death in the crushing pressures of Jupiter’s inner atmosphere (the pressure-distance relationship isn’t linear, by the way; beyond the Jovian upper atmosphere very little is known about its actual composition, so feel free to make up things as you go along) then nobody’s really going to be checking up on them anyway.[/quote]

Actually, they’re not going to be crushed. I wanted to know about Jupiter’s atmospheric pressure at different depths so I’d know how far they can enter it, and thus what initial velocity upon entering the atmosphere wouldn’t require fatal acceleration to prevent the ship from being crushed. I mean if they have to brush off 100 km/s in 1000 km, then they might as well swallow the suicide pills. But Q.E.D.'s link provides enough information to get an estimate on reasonable speeds. I wasn’t concerned about being precise, just something in the neighborhood of reality.

I don’t know how to integrate with respect to time when time isn’t in the equation. I guess that’s my problem. I know A = G * m / r^2 but getting time into that equation seems impossible. If it’s really that messy, I’ll just write a computer program that calculates the gravity every second or so of the trip and from that, the speed. It’s not perfect, but it should be close with small enough intervals.

9

It’s already there, in the constant G. It’s value is approximately 6.67300 × 10[sup]-11[/sup] m[sup]3[/sup] kg[sup]-1[/sup] s[sup]-2[/sup]. The “s” doesn’t stand for “salami”, y’know.

10

Okay, I just finished writing a program which I believe does just what I want. I have checked over my equations several times and can’t find any error. The algorithm works backwards assuming an initial velocity upon leaving the atmosphere and calculating the distance and velocity after a certain time. I calculated the initial velocity by assuming there had to be 0 velocity at 1100 km into the atmosphere (the point at which the ship has to stop or be crushed) and that there was an average of 40 m/s^2 (about 4 G’s) of acceleration. Since it won’t be a constant acceleration, that gives me a little breathing room I think since the body should be able to handle more than that. (I can research the body’s tolerance to g-force later. It’s not that important at this point.) The problem seems to be that it only takes about 6 1/2 minutes for the ship to start drifting back towards the planet. That’s not exactly very long to be stranded in space… I can’t even play with the numbers to get anything reasonable. It’s possible that I made a mistake in my calculations though, so can you guys check my work before I consider a totally different premise for the story?

Here’s how I calculated the velocity upon entering the atmosphere:



40m/s^2 = dv / t
dv = 2 * (1100000m / t)
dv = 2200000m / t
40m/s^2 = 2200000m / t^2
t^2 = 2200000m / 40m/s^2
t^2 = 55000s^2
t = 235s
iv = 235s * 40m/s^2
iv = 9400m/s

If it’s not obvious, dv is change in velocity, iv is initial velocity, and t is time, all in meters and seconds.

Here’s the line in my program to calculate the change in velocity for one second. Even if you don’t know C, you can probably figure out how it works.



v -= G * M / ((d + 71492000) * (d + 71492000));

G is the gravitational constant, M is the mass of Jupiter in kilograms, d is the distance from the surface of the planet, and 71,492,000 is the radius of the planet in meters. Is that all correct?

Well yeah, the units are there, but there’s no variable to solve for to get time.

11

If all you want is a velocity and you don’t care about position, forget Kepler or any calculus. Just use energy methods, i.e. the sum of your potential and kinetic energy is a constant. Figure your speed and gravitational potential from your initial position, and then figure your grav potential at whatever arbitrary distance, and that’ll give you your speed (but not vector velocity) from your kinetic energy.

What you have to take into consideation is the fact that your ship isn’t going to be sitting stationary in space, but rather in an elliptical orbit around the planet once they’ve escaped from certain death thanks to intrepidity of your Swiftian hero in devising a new propulsion system on the fly (don’t shatter my illusions here) and so will be becalmed with no way to effect rescue or hail someone.

You might want to rethink the entry into the atmosphere, though; at orbital speeds even a very low pressure will create substantial drag and (unless your hull is made out of Puppeteer material or the unobtainium that plates the hulls of Serenity) will heat up and lose strength or erode away rapidly upon re-entry.

Stranger