Limb brightening

[ericagol]

Do we want to include a limb-brightening term, such as that in Schlawin et al. http://adsabs.harvard.edu/abs/2010ApJ...722L..75S

[rodluger]

Yes, that's a good idea. But the math is no different, right? We just switch the sign on the spherical harmonic coefficients to get a profile that brightens towards the limb.

[ericagol]

It actually is different - the surface brightness (formally) becomes infinite at the limb due to the cusp caused by the tangent through the atmosphere.

[rodluger]

Interesting. What is the functional form of the profile then? It looks like Schlawin's paper only computes the analytic expression for the thin shell approximation, right? Perhaps we can use Green's theorem to get a better solution?

On Tue, Apr 24, 2018 at 12:03 PM Eric Agol notifications@github.com wrote:

It actually is different - the surface brightness (formally) becomes infinite at the limb due to the cusp caused by the tangent through the atmosphere.

— You are receiving this because you commented.

Reply to this email directly, view it on GitHub https://github.com/rodluger/limbdark/issues/4#issuecomment-383966457, or mute the thread https://github.com/notifications/unsubscribe-auth/AI5FK3IGLCX14Y_rIhT674TumwJKzYq2ks5trz7UgaJpZM4TgrFO .

[ericagol]

Yes, just the thin shell approximation. Green’s theorem is a good idea.

Eric Agol Astronomy Professor University of Washington

On Apr 24, 2018, at 8:32 AM, Rodrigo Luger notifications@github.com wrote:

Interesting. What is the functional form of the profile then? It looks like Schlawin's paper only computes the analytic expression for the thin shell approximation, right? Perhaps we can use Green's theorem to get a better solution?

On Tue, Apr 24, 2018 at 12:03 PM Eric Agol notifications@github.com wrote:

It actually is different - the surface brightness (formally) becomes infinite at the limb due to the cusp caused by the tangent through the atmosphere.

— You are receiving this because you commented.

Reply to this email directly, view it on GitHub https://github.com/rodluger/limbdark/issues/4#issuecomment-383966457, or mute the thread https://github.com/notifications/unsubscribe-auth/AI5FK3IGLCX14Y_rIhT674TumwJKzYq2ks5trz7UgaJpZM4TgrFO .

— You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub, or mute the thread.

[rodluger]

Are there any functions that tend to infinity as x --> 1 but are integrable over the range [0, 1]?

[ericagol]

Yes: 1/sqrt(1-x)

Eric Agol Astronomy Professor University of Washington

On Apr 24, 2018, at 10:25 AM, Rodrigo Luger notifications@github.com wrote:

Are there any functions that tend to infinity as $x --> 1$ but are integrable over the range [0, 1]?

— You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub, or mute the thread.

[rodluger]

Cool. I played around with some derivatives and this is what I got. If our limb-brightening profile is

I(mu) = 1 / mu

where

mu = sqrt(1 - x^2 - y^2)

then the brightness tends to infinity at the limb (x^2 + y^2 --> 1), but the integral over the disk still converges (it's equal to 2*pi).

After some trial and error, I found that the exterior derivative of the vector function

G = {y * mu(x, y), -x * mu(x, y)}

is

D ^ G = 1 / mu - 3 * mu

This is almost what we want. If we can compute the line integral of G, then I think we can just add the line integral corresponding to the 3 * mu term, which is just the linear limb darkening solution, to get the occultation flux for the limb-brightening solution!

[ericagol]

Interesting - perhaps any \mu^\alpha works with \mu positive or negative. One nice thing about the model in Schlawin et al. is that it does have a physical motivation: if a star has optically-thin, uniform emission, then the intensity is well approximated by the surface of a sphere, which is what that model computes.

[rodluger]

That's a neat idea. The spherical harmonics cover \mu^\alpha for positive \alpha, and this basis would cover it for negative \alpha to get something like a Laurent series to model arbitrary limb brightening/darkening.