Re: Gravity of a Torus

Logan Kearsley wrote:

Except that when I found that I could not solve the one parameterized in
theta, I wondered if maybe parameterizing it in terms of x would get me
something solvable (it didn't).

Oh, I see. You tried it one way, then tried to formulate it another way and tried again. I thought you were saying you were trying both simultaneously as in a double integral.

If I didn't screw up my algebra anywhere, the correct expression to replace
dtheta is arcsin(x)dx, and that gives me another unsolvable integral. I
suppose I shall see what happens if I plug in the taylor series for arcsin
next.

The first two terms you end up with are well-known forms called elliptical integrals, which do not have closed-form solutions.

Only if I know where those terms come from. The page I linked to doesn't
give any derivation, so I have very little idea what terms to add.

Since they're infinite series, I'm sure they're just Taylor series approximations (as you already guessed I believe).

So, in short, there really is no analytic solution for the integral? Darn.
Oh well, I'll be happy with an infinite series, If I can find out what the
series is.

What I really want is to be able to integrate the field of a torus to get
the field of an annulus, so as to be able to model a flat planet and/or
Alderson disk.

_This_, by the way, _would_ involve a double integral. This kind of integral, actually, is how you find the gravitational field above an infinite, uniform plane. (The answer is that the field is constant everywhere.)

Thanks! One question, though: doesn't one have to add a factor to project
the field from each mass element into the x-z plane, since opposite halves
of the torus balance in the y-axis?

No, the general formulation accounts for all that. Or, to put it another way, by symmetry arguments you know that the second term has to vanish.

--
Erik Max Francis && max@xxxxxxxxxxx && http://www.alcyone.com/max/
San Jose, CA, USA && 37 20 N 121 53 W && AIM erikmaxfrancis
In this world, nothing is certain but death and taxes.
-- Benjamin Franklin
.

Relevant Pages

  • Re: Integral derivation ...
    ... their calculators and check whether equality holds, ... Just evaluating the two integrals ... can be expressed as infinite series in b. ...
    (comp.dsp)
  • Re: x+sin x =2 implies x=?
    ... of the functions you'd come across in 1st year Calculus, ... And if we do not stop at infinite series (and infinite products, ... parametric integrals), we might consider infinite compositions, such as ...
    (sci.math)
  • Re: Integral of exp(sin(x)) dx
    ... is there a compelling reason why such integrals CANNOT be ... infinite series but in that sense they are not elementary). ... still think of integrals of these trig functions as elementary because they ... are so common and do have special properties that makes them easy to work ...
    (sci.math)
  • Re: Summation over non-integers
    ... Euler-Maclaurin summation formula, and treat the sum ... terms fas the corresponding integrals plus some ... necessarily true in the general case), or, I suppose, ... An infinite series, at least, ...
    (sci.math)