Calculate ionizing photon rate from black body spectrum

[will-henney]

The PION simulation code uses a black body approximation for the spectrum of the central star when calculating the ionization of hydrogen. The input parameters that we can vary are the effective temperature and radius of the star.

However, we want to be able to explicitly control the ionizing luminosity $QH$ so that we can compare with analytic models. The task for @RobeReyes is to calculate the hydrogen ionizing luminosity of a black body source with radius $R\star$ and effective temperature $T_\star$.

In particular, for the WR124 parameters that we have been using:

RT_Rstar____0    18.0   # Radius of star in solar radii.
RT_Tstar____0    3.3e4     # Stellar effective temperature.

I suggest using astropy.units to help ensure that units are correct. You can either write your own Planck function routine or use astropy.modelling.models.BlackBody(). The integration can be done with methods in scipy.integrate

[will-henney]

@RobeReyes please respond to this thread with an explanation of what equations you will be using before starting work on the programming

[will-henney]

We can test the results by comparing with this graph, which comes from Leitherer et al 1990

Note that these results are for the little $q$, which is surface flux instead of lumihosity, so it doe snot include the factor of $4 \pi R_*^2$.

and the $F\nu$ here is the astrophysical flux rather than the physical flux, so it is numerically equal to our $B\nu$

[will-henney]

Here are the notes that I took during the class on Wednesday:

[RobeReyes]

In my archive i have the same but the factor 2 i put out of the integral

[will-henney]

I found that we can make an approximation that exp(h nu / kT) >> 1, which is good to about 1% in the temperature range that we are interested in. This means that we can do the integral analytically, which makes things much easier!

Dr William Henney, Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Campus Morelia

On 1 Sep 2023 13:14:44, RobeReyes @.***> wrote:

In my archive i have the same but the factor 2 i put out of the integral

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[will-henney]

Final result of this is given here:

https://github.com/will-henney/globule-seminario/blob/main/m1-67/ionizing-luminosity.org

The result is $1.2 \times 10^{49}$ per second for $T\mathrm{eff} = 33,000$ K and $R = 18 R\odot$. This is very close to the result fro WR model atmosphere models