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I would like to know if an exact solution for the surface gravity force components of an oblate spheroid has been published and if not can anyone derive it here?

Assume an ideal rigid oblate sphere of uniform homogenous density, that is not rotating. We can always add in centripetal forces later if required.

The place to start would seem to be this exact general solution for the gravitational potential:

$$P = \frac{3 M}{(2 a e)} \left[B \left(1 - \frac{r^2}{(2 a^2 e^2)} + \frac{z^2}{(a^2 e^2}\right) + \frac{r^2}{(2 a^2 e^2)} \sin(B) \ \cos(B) - \frac{z^2}{(2 a^2 e^2)} \tan(B) \right] \ \ $$

where $$B = \frac 1 2 \arccos \left( \sqrt{\frac{4 a^2 e^2}{r^2} + \left(1+\frac{a^2 e^2 + z^2}{r^2}\right)^2} - \ (a^2 e^2 + z^2)/r^2 \right),$$

$M$ is the mass, $a$ is the semi-major axis of the elliptical cross section of the oblate, "e" is the eccentricity of the oblate, $z$ is the vertical distance from the equatorial plane and $r$ is the horizontal distance from the vertical axis of rotational symmetry, of the point on the surface of the oblate.

This equation for the external potential of an oblate spheroid of homogenous density was derived by Gauss and Dirichlet. See this 2018 paper by Hofmeister et al that was linked to by @ProfRob in a question about the gravity of an oblate sphere in the Astronomy forum.

I do not want solutions that depend on $J_2$ harmonic approximations as these are only generally valid for low eccentricity bodies.

I would like to know if an exact solution for the surface gravity force components of an oblate spheroid has been published and if not can anyone derive it here?

Assume an ideal rigid oblate sphere of uniform homogenous density, that is not rotating. We can always add in centripetal forces later if required.

The place to start would seem to be this exact general solution for the gravitational potential:

$$P = \frac{3 M}{(2 a e)} \left[B \left(1 - \frac{r^2}{(2 a^2 e^2)} + \frac{z^2}{(a^2 e^2}\right) + \frac{r^2}{(2 a^2 e^2)} \sin(B) \ \cos(B) - \frac{z^2}{(2 a^2 e^2)} \tan(B) \right] \ \ $$

where $$B = \frac 1 2 \arccos \left( \sqrt{\frac{4 a^2 e^2}{r^2} + \left(1+\frac{a^2 e^2 + z^2}{r^2}\right)^2} - \ (a^2 e^2 + z^2)/r^2 \right),$$

$M$ is the mass, $a$ is the semi-major axis of the elliptical cross section of the oblate, "e" is the eccentricity of the oblate, $z$ is the vertical distance from the equatorial plane and $r$ is the horizontal distance from the vertical axis of rotational symmetry, of the point on the surface of the oblate.

This equation for the external potential of an oblate spheroid of homogenous density was derived by Gauss and Dirichlet. See this 2018 paper by Hofmeister et al that was linked to by @ProfRob in a question about the gravity of an oblate sphere in the Astronomy forum.

I do not want solutions that depend on $J_2$ harmonic approximations as these are only generally valid for low eccentricity bodies. I am looking for exact general solutions that can be safely applied to bodies with high eccentricity

I would like to know if an exact solution for the surface gravity vector components of an oblate spheroid has been published and if not can anyone derive it here?

Assume an ideal rigid oblate sphere of uniform homogenous density, that is not rotating. We can always add in centripetal forces later if required.

The place to start would seem to be this exact general solution for the gravitational potential:

$$P = \frac{3 M}{(2 a e)} \left[B \left(1 - \frac{r^2}{(2 a^2 e^2)} + \frac{z^2}{(a^2 e^2}\right) + \frac{r^2}{(2 a^2 e^2)} \sin(B) \ \cos(B) - \frac{z^2}{(2 a^2 e^2)} \tan(B) \right] \ \ $$

where $$B = \frac 1 2 \arccos \left( \sqrt{\frac{4 a^2 e^2}{r^2} + \left(1+\frac{a^2 e^2 + z^2}{r^2}\right)^2} - \ (a^2 e^2 + z^2)/r^2 \right),$$

$M$ is the mass, $a$ is the semi-major axis of the elliptical cross section of the oblate, "e" is the eccentricity of the oblate, $z$ is the vertical distance from the equatorial plane and $r$ is the horizontal distance from the vertical axis of rotational symmetry, of the point on the surface of the oblate.

This equation for the external potential of an oblate spheroid of homogenous density was derived by Gauss and Dirichlet. See this 2018 paper by Hofmeister et al that was linked to by @ProfRob in a question about the gravity of an oblate sphere in the Astronomy forum.

I do not want solutions that depend on $J_2$ harmonic approximations as these are only generally valid for low eccentricity bodies.

I would like to know if an exact solution for the surface gravity force components of an oblate spheroid has been published and if not can anyone derive it here?

Assume an ideal rigid oblate sphere of uniform homogenous density, that is not rotating. We can always add in centripetal forces later if required.

The place to start would seem to be this exact general solution for the gravitational potential:

$$P = \frac{3 M}{(2 a e)} \left[B \left(1 - \frac{r^2}{(2 a^2 e^2)} + \frac{z^2}{(a^2 e^2}\right) + \frac{r^2}{(2 a^2 e^2)} \sin(B) \ \cos(B) - \frac{z^2}{(2 a^2 e^2)} \tan(B) \right] \ \ $$

where $$B = \frac 1 2 \arccos \left( \sqrt{\frac{4 a^2 e^2}{r^2} + \left(1+\frac{a^2 e^2 + z^2}{r^2}\right)^2} - \ (a^2 e^2 + z^2)/r^2 \right),$$

$M$ is the mass, $a$ is the semi-major axis of the elliptical cross section of the oblate, "e" is the eccentricity of the oblate, $z$ is the vertical distance from the equatorial plane and $r$ is the horizontal distance from the vertical axis of rotational symmetry, of the point on the surface of the oblate.

This equation for the external potential of an oblate spheroid of homogenous density was derived by Gauss and Dirichlet. See this 2018 paper by Hofmeister et al that was linked to by @ProfRob in a question about the gravity of an oblate sphere in the Astronomy forum.

I do not want solutions that depend on $J_2$ harmonic approximations as these are only generally valid for low eccentricity bodies.

I would like to know if an exact solution for the surface gravity vector components of an oblate spheroid has been published and if not can anyone derive it here?

Assume an ideal rigid oblate sphere of uniform homogenous density, that is not rotating. We can always add in centripetal forces later if required.

The place to start would seem to be this exact general solution for the gravitational potential:

$$P = \frac{3 M}{(2 a e)} \left[B \left(1 - \frac{r^2}{(2 a^2 e^2)} + \frac{z^2}{(a^2 e^2}\right) + \frac{r^2}{(2 a^2 e^2)} \sin(B) \ \cos(B) - \frac{z^2}{(2 a^2 e^2)} \tan(B) \right] \ \ $$

where $$B = \frac 1 2 \arccos \left( \sqrt{\frac{4 a^2 e^2}{r^2} + \left(1+\frac{a^2 e^2 + z^2}{r^2}\right)^2} - \ (a^2 e^2 + z^2)/r^2 \right),$$

$M$ is the mass, $a$ is the semi-major axis of the elliptical cross section of the oblate, "e" is the eccentricity of the oblate, $z$ is the vertical distance from the equatorial plane and $r$ is the horizontal distance from the vertical rotation axis of the point on the surface of the oblate.

This equation for the external potential of an oblate spheroid of homogenous density was derived by Gauss and Dirichlet. See this 2018 paper by Hofmeister et al that was linked to by @ProfRob in a question about the gravity of an oblate sphere in the Astronomy forum.

I do not want solutions that depend on $J_2$ harmonic approximations as these are only generally valid for low eccentricity bodies.

I would like to know if an exact solution for the surface gravity vector components of an oblate spheroid has been published and if not can anyone derive it here?

Assume an ideal rigid oblate sphere of uniform homogenous density, that is not rotating. We can always add in centripetal forces later if required.

The place to start would seem to be this exact general solution for the gravitational potential:

$$P = \frac{3 M}{(2 a e)} \left[B \left(1 - \frac{r^2}{(2 a^2 e^2)} + \frac{z^2}{(a^2 e^2}\right) + \frac{r^2}{(2 a^2 e^2)} \sin(B) \ \cos(B) - \frac{z^2}{(2 a^2 e^2)} \tan(B) \right] \ \ $$

where $$B = \frac 1 2 \arccos \left( \sqrt{\frac{4 a^2 e^2}{r^2} + \left(1+\frac{a^2 e^2 + z^2}{r^2}\right)^2} - \ (a^2 e^2 + z^2)/r^2 \right),$$

$M$ is the mass, $a$ is the semi-major axis of the elliptical cross section of the oblate, "e" is the eccentricity of the oblate, $z$ is the vertical distance from the equatorial plane and $r$ is the horizontal distance from the vertical axis of rotational symmetry, of the point on the surface of the oblate.

This equation for the external potential of an oblate spheroid of homogenous density was derived by Gauss and Dirichlet. See this 2018 paper by Hofmeister et al that was linked to by @ProfRob in a question about the gravity of an oblate sphere in the Astronomy forum.

I do not want solutions that depend on $J_2$ harmonic approximations as these are only generally valid for low eccentricity bodies.