Homework 2

Note

You are free to discuss these questions with your classmates and on our class slack, but you must write your own solutions, including your own source code.

All code should be uploaded to Brightspace along with any analytic derivations, notes, etc.

1. Stellar colors are expressed as the difference in astronomical magnitudes with the flux measured through different filters.

   The standard photometric system defines U, B, V, and more filters, where U is ultraviolet, B is blue, and V from which we can define a color index like \(B-V\) (where here, \(B\) is now the magnitude of the star through the B filter). It is \(B-V\) that traditionally appears as the \(x\)-axis in the Hertzsprung-Russel diagram.

   \(B-V\) is a difference in magnitudes, which can be expressed as:

   \[B - V = -2.5 \log_{10}(f_B/f_V)\]

   where \(f_B\) is the flux received from a star through the B filter and \(f_V\) is the flux from a star through the V filter.

   We can treat a star as a perfect blackbody (narrator: they are not), and then take

   \[f_X \approx \int_0^\infty B_\lambda(T) s_X(\lambda) d\lambda\]

   where \(B_\lambda(T)\) is the Planck function for temperature \(T\), and \(s_X\) is the filter X’s response (I know, too many things are getting called B…). The “\(\approx\)” here is because the distance to the star, its radius, and the fact that we need to integrate the Planck function over the outward angles all introduce a proportionality constant. However, since we are taking the ratio of fluxes, this proportionality constant will cancel.

   We will take the filter to be a box-filter, with:

   \[\begin{split}s_X(\lambda) = \left \{ \begin{array}{cc} 1 & |\lambda - \lambda_{0,X}| \le \frac{d\lambda_X}{2} \\ 0 & \mbox{otherwise} \end{array} \right . \end{split}\]

   (i.e., the filter only lets light through in a narrow band of width \(d\lambda_X\) centered on a wavelength \(\lambda_{0,X}\).) Reasonable values for the filter properties are:

   filter	\(\lambda_{0,X}\)	\(d\lambda_{X}\)
   B	445 nm	90 nm
   V	550 nm	90 nm

   There is one further complication. Magnitudes have a zero-point (calibration). It used to be that Vega was taken to have a magnitude of 0 in all filters (but Vega is variable, so a different procedure is used now). We will assume something Vega-like, and define \(B - V = 0\) for \(T = 10000~\mathrm{K}\). This means that you will need to shift your results by a constant offset.

   Your task

   Using either the trapezoid rule or Simpson’s rule, write a code that computes \(B-V\) for 20 temperatures evenly spaced in \(\log_{10}(T)\) between 4000 K and 25000 K, and plot \(B-V\) vs. \(T\).

   Make sure you demonstrate that you used enough bins in your integral (perhaps by doubling the number of bins and showing that the change in the answer is small).

   The add the following data points of \(B-V\) from real stars (data taken from Carroll & Ostlie):

   spectral type	\(T\)	\(B - V\)
   B0	30000	-0.30
   A0	9520	-0.02
   F0	7200	+0.30
   G0	6030	+0.58
   K0	5250	+0.81
   M0	3850	+1.40