$\alpha*r$" />I am having trouble comparing the 2-dimensional gravitational flux, due to a point mass $M$ located at the origin of the gaussian circle (in green) at point O, versus a point mass located at some distance from the origin given by $\alpha$r, similar to (but with a different approach) another unanswered post about offset electric charge: Gauss's Law for an offset (non-origin) charge. I figured from what I've read, the flux for both the center/offset cases is supposed to be the same, which makes sense: Why does Gauss's law work for a charge off center in a spherical surface?. However, I am not getting the same flux for both these cases, and I'm not sure if its something to do with the physics or just something wrong with my calculations.
Now, the previous two links were regarding charges in 3D space, but I presumed that the reasoning followed for newtonian gravity in 2D space, albeit with a $\frac{1}{r}$ dependence and not a $\frac{1}{r^2}$ (Newtonian gravity equation in a 2 dimensional world.
In the image, I placed a point mass $M$ at point A, located at a distance of $\alpha$r from the origin O where $\alpha$ is between 0 and 1. The gravitational field from this mass at test point T would then be given as: $$\vec{F}=\frac{G*M}{(AT)}*\hat{a_t}$$ where AT is the length of the line segment connecting point A and T, and $\hat{a_t}$ is the unit vector pointing towards the point mass. The length of AT would be a function of angle $\theta$ with $\theta=0^{\circ}$ corresponding to line segment OT in line with AT. I derived AT as: $$AT=r\sqrt{1+\alpha(\alpha-2cos(\theta))}$$ through the use of the cosine law. I derived the angle between the radially directed unit vector $\hat{a_r}$ and the unit vector $\hat{a_t}$ to be: $$cos(180-\epsilon)=-cos(\epsilon)=-\sqrt{1-\frac{\alpha^2sin^2(\theta)}{1+\alpha(\alpha-2cos(\theta))}}$$ where $\epsilon$ is the angle between line segments AT and OT. $cos(\epsilon)$ was found by first deriving $sin(\epsilon)$ through the sine law, and then manipulating. Given the above two equations for AT and cos(180-$\epsilon$), I derive the flux crossing the circular boundary as follows: $$\phi_{offset}=\int_{0}^{\pi} \frac{2*G*M}{AT}*\hat{a_t} *dl*\hat{a_r}=\int_{0}^{\pi} \frac{2*G*M}{r\sqrt{1+\alpha(\alpha-2cos(\theta))}}*rd\theta*\hat{a_t} *\hat{a_r}$$ $$\phi_{offset}=\int_{0}^{\pi} \frac{-2*G*M}{\sqrt{1+\alpha(\alpha-2cos(\theta))}}*cos(\epsilon)*d\theta.$$ The integral is taken from 0 to $\pi$ since the cos/sin laws are only valid from 0 to $\pi$. The flux for a regular origin centered mass is given by: $$\phi_{center}=\int_{0}^{\pi}\frac{2*G*M}{r}*(-\hat{a_r})*rd\theta*\hat{a_r}=\int_{0}^{\pi}-2*G*M*d\theta.$$ To equal the integral bounds for $\phi_{offset}$, the integrand is taken from 0 to $\pi$ which adds a factor of 2. For the two fluxes to be equal, i.e. $\phi_{offset}=\phi_{center}$, $$\frac{-2*G*M}{\sqrt{1+\alpha(\alpha-2cos(\theta))}}*\sqrt{1-\frac{\alpha^2sin^2(\theta)}{1+\alpha(\alpha-2cos(\theta))}}=-2*G*M$$ Simplifying a little gives, $$\sqrt{1-\frac{\alpha^2sin^2(\theta)}{1+\alpha(\alpha-2cos(\theta))}}=\sqrt{1+\alpha(\alpha-2cos(\theta))}.$$ I've plotted the left and right side of the equation given above, but these don't match except for at $\alpha=0$. The process I followed makes sense to me, and I don't see any apparent errors in my calculation. The only thing I can think off is that my flux equations are somehow off.
I know there might be easier ways to prove this, and I would love to hear them! But I would also like to know why what I'm doing isn't working for my own sanity. Also sorry if I missed anything; this is my first post so I tried to cram as much detail as possible!
