Equal area projection of a sphere

I'm trying to create an equal area projection from a sphere. The sphere might not necessarily be the Earth. I would like, for instance, to use the Sun. Using the code

GeoGraphics[
GeoRange -> All,
GeoModel -> Entity["Star", "Sun"],
GeoProjection -> "Orthographic"]

doesn't work. This must have something to do with the data that Mathematica has for the Sun.

So, I decided that I need to create my own sphere like so:

SphericalPlot3D[1, {u, 0, Pi}, {v, 0, 2 Pi},
PlotPoints -> 50,
TextureCoordinateFunction -> ({#5, 1 - #4} &),
PlotStyle -> Directive[Texture[Import["https://i.stack.imgur.com/YOzSq.jpg"]]]] Now, I want to use an equal area projection. Preferably, I would like to use GeoProjection and GeoProjectionData. But I am having trouble putting it into the correct format. I don't need to use GeoProjection, but it seems like a good option.

ImageTransformation[] is useful for this. I assume centering at $(\varphi,\lambda)=(0,0)$; for other centers, I'll leave the adjustment to you.

sun = Import["https://i.stack.imgur.com/YOzSq.jpg"];

ImageTransformation[sun, If[#[]^2 + #[]^2 < 1,
{Arg[Sqrt[1 - #[]^2 - #[]^2] + I #[]],
ArcSin[#[]]}, {π, π}] &,
DataRange -> {{-π, π}, {-π/2, π/2}}, Padding -> 1.,
PlotRange -> {{-1, 1}, {-1, 1}}] • (For comparison's sake, replace sun in the above code with your favorite equirectangular map of Earth, and compare with GeoGraphics[GeoRange -> All, GeoProjection -> "Orthographic"].) – J. M. will be back soon Jan 24 '17 at 12:59