u/Xovvo

I suddenly realized that I don't understand functions in spherical coordinates, nor, evidently, how circles work. What I want to do is tilt a circle (centered on the origin, in the xy-plane) in 3D space. In Cartesian coordinates it kinda sucks, but is doable, you rotate the plane intersecting the sphere that defines the center and radius by the angle you want. Taking that resulting equation into spherical coordinates gives me either 1 (if I include a z-term), or r = sqrt(cos(θ)^2 + sin(θ)^2 * cos(ɑ)^2) which gives me a volume bc of course it does, but I notice that this volume kinda looks like if I take the right slice of it, I'd get the circle I want.

I've tried approaching this from the other direction, letting θ vary freely, with r=R, and having φ=pi/2 + Xcos(θ)---but I can't quite figure out what the X should be.

Idk, I feel stupid, bc it can't be this hard to define a function that traces a circle of arbitrary orientation in spherical coordinates---not even an arbitrary circle, just getting a vector to trace a great circle of arbitrary polar angle!
But I can't get it to work and it's driving me mad. What is it that I'm failing to understand?

I'm following along with Fitzpatrick's calculation of the precession of Earth's axis caused by the Sun tugging on Earth's equatorial bulge, here. I've calculated the Gravitational potential outside a uniform oblate spheroid (out to the Quadrupole term) to be:

φ here is the polar angle of our position vector of magnitude r

Or, if you insist as Fitzpatrick does (for valid reasons, tbh) on having things in terms of Legendre polynomial terms:

φ here is the polar angle of our position vector of magnitude r

Great! It should be noted that the major radius a is the equatorial radius of the massive body, not the satellite.

When calculating the potential energy of the Earth, Fitzpatrick is clearly multiplying the gravitational potential generated by the Sun at Earth by the mass of the Earth---which is a little weird bc we want the torque on its equatorial bulge, but I'm willing to trust Fitzpatrick here---but then he decides to use MacCullagh's formula to bundle one-fifth the mass of the earth multiplied by the product of the square of the equatorial radius and the square of the eccentricity into the difference between the major and minor moments of inertia of the Earth.

gamma sub s is the angle between the position vector of the sun and the angular velocity vector of the earth

Except that "a" here isn't the equatorial radius of the Earth, it's the Equatorial radius of the Sun, so we can't actually do this, but here it is done anyway.

This all feels weird. How should I actually be deriving the torque exerted on the bulge of the Earth from the potential generated by the Sun at Earth over the course of a year?

I have been trying to calculate the quadrupole component of the gravitational potential of an oblate spheroid, and I had thought I had a handle on it, but when reading through Fitzpatrick's Newtonian Dynamics in the section on the potential outside a uniform spheroid I noticed that the quadrupole component had a coefficient of 2/5 as opposed to the 3/10 I obtained. They use a very weird boundary function for the spheroid (different from both mine and the one found here for the radial distance from the rotation axis; I don't know how Fitzpatrick is getting away with not dividing by sine or cosine), and they make a "simplifying" assumption based on the axial symmetry of the mass distribution, allowing them to replace the cosine of the azimuthal angle and the square of the cosine of the azimuthal angle with 0 and 1, respectively, and the integral over the azimuthal angle with twice pi, but that shouldn't mean our answers are so different?

In any case, I had a hard time getting the same answer twice (though I managed to do exactly that when typesetting my work in LaTeX), so I would like to confirm that I have the correct result for the gravitational potential.

so

https://preview.redd.it/gxcqzog2bx3h1.png?width=583&format=png&auto=webp&s=a4718a85b0f0d7961fb6e5aedca9f49440511273

Defining some functions that will let us save space and effort later:

https://preview.redd.it/5ovu44h6bx3h1.png?width=580&format=png&auto=webp&s=ccb01cdaaf62b43de7299daeab304de2cbdc0856

A selection of derivatives and integrals we'll be making use of:

the power rule is not listed, but maybe it should be, for completeness?

This term will come up, so let's go ahead and calculate it now:

https://preview.redd.it/5thlujucbx3h1.png?width=501&format=png&auto=webp&s=a102f7d7706f592db44356825badf1420597f7c1

The square of this term will come up, so lets go ahead and calculate it:

https://preview.redd.it/f3rhfjwfbx3h1.png?width=795&format=png&auto=webp&s=11d6f97d49dd8be5c4ee33e010d087beabb57404

Obtaining the boundary function for the radius:

I'm pretty sure the denominator is never zero, so long as a and c are not zero.

Verifying the monopole term:

derivation of the volume from a triple integral in spherical coordinates omitted here, for secret reasons.

Moving on to the Quadrupole term:

I'm so sorry

Breaking each term into it's own integral, and "factoring out" the theta prime terms:

https://preview.redd.it/uuu8qp4hcx3h1.png?width=582&format=png&auto=webp&s=3c630db4fff87d6a3e27a9cea01e4896e1efd62a

Solving the integrals of theta prime:

Now our big integral is three smaller integrals we will treat separately.

Working on the first sub-integral:

https://preview.redd.it/9vde7abucx3h1.png?width=578&format=png&auto=webp&s=61d6419a3d7be87d1331f0a8494f4f33484715ca

Multiplying through and applying the integrals in Eq. 11 & 12:

Please ignore the typesetting error cutting of a closing parenthesis.

and so

Success?

Now we move on to Integral B:

https://preview.redd.it/809rtliddx3h1.png?width=585&format=png&auto=webp&s=2314e7cfc5c8e0de66d989eca14cbe6308fabbf7

applying the integral in Eq. 12:

Success?

Now on to Integral C:

https://preview.redd.it/wqiadhjpdx3h1.png?width=585&format=png&auto=webp&s=dbdc5d7d301fd823a7c2bb4245d01682ce7f6ef0

applying the integral in Eq. 11:

Success?

Now we put it all together and resolve:

Final answer

...and that's what I get. Did I mess something up somewhere?

I'm working through the "Gravitational Potential Theory" section of Newtonian Dynamics by Richard Fitzpatrick and hosted here on the University of Texas at Austin website, for personal reasons; I got to the section on "McCullough's Formula" (also rendered "MacCullagh's Formula"), and it claims the "ellipticity" can be related to the difference between the moment of inertia around the axis of rotation and the moment of inertia perpendicular to that axis, divided by the moment of inertia of a sphere of mean radius---and homotopically equal to that same difference, divided by the moment of inertia around the axis of rotation.

I wanted to confirm that, especially since nothing in the text that I could find could clarify whether they meant the eccentricity (ɛ = sqrt(1-(c/a)^2) or the flattening
(also called ellipticity or oblateness; f = (a-c)/a), so I tried plugging in the moments of inertia, and making use of the mean-radius relation r = (2a+c)/3 then for oblate spheroids a=b > c:

I_|| = (1/5)M(2a^2)
I_⟂ = (1/5)M(a^2 + c^2)
I_o = (2/5)M((2a+c)/3)^2

I_|| - I_⟂ = (1/5)M(a^2 - c^2)

(I_|| - I_⟂)/I_o = (9/2)((a^2 - c^2)/(2a + c)^2)

(I_|| - I_⟂)/I_|| = (1/2)(1 - (c/a)^2) = (1/2) ɛ^2

Neither of which are the eccentricity or the flattening. I must be missing something, but I don't know what that is?

While trying to confirm I had the right limits of integration and function for the volume of an oblate spheroid (with this sub's corrections on where I went wrong the first set of attempts), I ran into a problem that I had no idea how to approach and had to resort to Wolfram Alpha to advance, and I would like to know what I should have done instead, since integration by parts gave me hard-to -resolve inverse trig functions, partial fraction decomposition wasn't appropriate for the task, and trying to do u-substitution a second time had me dividing by u, which IIRC is Not Allowed bc that's doing a lot more than just substitution.

Here is the problem set up:

setting up the limit of integration for rho

Moving on to the volume integral:

Equation 17 is the limit of my skill

And here I hit a wall. It kinda looks like the second derivative of the inverse hyperbolic sine function, but it's not quite the right shape, attempting to do u-substitution as w = c^2 + (a^2 - c^2)u^2 leaves us with dw = 2(a^2 - c^2)u du which we don't have available. Trig substitution feels weird to use here since I used u-substitution to get rid of inconvenient trig functions. I'm unsure what to do here, but Wolfram Alpha had no trouble with integrals of the form (c^2 + (a^2 - c^2)u^2)^(-3/2):

But after obtaining Equation 18, I have the skills to resolve the integral.

What was it that I should have done?

I'm trying to obtain the volume of an oblate spheroid through a triple integral in spherical coordinates. I'm starting with the general equation for an oblate spheroid and taking that into polar coordinates, solving for the radius in terms of the angle phi, and then integrating with respect to angle theta, angle phi, and radius rho---but I can't seem to get it to come out to the correct (4*pi/3)*c*a^2. I'm not sure what I'm doing wrong, though, typing this out I'm wondering if starting with the equation of a surface kind of precludes a triple integral for finding the volume, but if not that equation, I'm not sure where I would start.

In any case, here's my work, please, see if you can find where I'm going wrong:

I forgot to add the identities to convert to spherical coordinates, but did include eccentricity in terms of the polar and equatorial radii, and the derivative of arcsinh(x) because that gets used later

Getting the radius as a function of angle phi:

https://preview.redd.it/4m8fo4zjlvxg1.png?width=583&format=png&auto=webp&s=6fdb84f52925dd7ad15b75057d52dbae7d570e4f

And now trying and failing to get the volume:

https://preview.redd.it/2jimwthwlvxg1.png?width=555&format=png&auto=webp&s=faa5e6c7bbd09fac50e877432841d4231e5680b1

making a substitution to continue working:

https://preview.redd.it/1r21t253mvxg1.png?width=601&format=png&auto=webp&s=247cb58bc171647fc3b244a6bcec9bd5bf5f4b1e