$2$ Coding Assignment to be complete in a jupiterlab notbook in python: Finding local maxima.

Throughout this assignment, you must comment your code and be sure to label the axes on all your plots. Please indicate clearly at the start of a block of code which part of the assignment it is in response to$.$

$1.$ Plot the Planck function for a temperature of T $=5800$ K $($you may use the function you defined in previous tutorials, either by pasting the code into your notebook or $($after Tuesday$$s $1$ class$)$ by importing it as a module$).$ Use a range of wavelengths $($plotted in nm$)$ that includes the peak of the thermal emission spectrum and sample the curve with enough points that it looks smooth.

$2.$ Compute the derivative of the Planck function dB$/$d over your range of wavelengths using a central difference scheme: f $0($x$0)=$ f$($x$0+$ h$)$ f$($x$0$ h$)2$h Using a step size of h $=0\mathrm{.}1$ nm$,$ evaluate this derivative at exactly the same wavelengths as before, and plot it on a new set of axes.

$3.$ Estimate visually the wavelength where the derivative dB$/$d crosses zero $($it may help to plot a straight line at y $=0$ using plt$.$axhline$).$ Does it correspond to the wavelength where the emission spectrum peaks? Should it$?$

$4.$ Plot the Planck function you plotted in Q$1.$ This time, color the points or curve so they are one color when the slope $($dB$/$d$)$ is positive and another color when the slope is negative. Hint: use Boolean selection arrays! Think carefully about how to use plt$.$plot$()$ or plt$.$scatter$()$ to accomplish this.

$5.$ Write code to find the wavelength location of the spectrum$$s peak, peak by computing where the derivative $($as calculated via central difference formula above$)$ changes sign from positive to negative $($this method is more robust than looking for an exact zero$).$ Do the same for $9$ stars with temperatures spaced every $1000$ K between $2000$ and $10000$ K$,$ inclusive. Plot peak as a function of temperature, and overplot the analytical result expected from Wien$$s Law $($Homework B$)$ on the same axes. Do your results $($roughly$)$ agree?

$6.$ Load the data in the file model$\_5800$K$.$txt $($available on both Canvas and Scorpius under $/$home$/$jdarling$/$astr$2600/$textfiles$/)$ into two arrays, one for wavelengths and the other for fluxes. This is a model spectrum of a Sun$-$like star. Plot this spectrum with wavelength on the x$-$axis and flux on the y$-$axis. On the same axes, overplot the Planck function for a $5800$K object.

$7.$ Calculate the derivative of this model spectrum with respect to wavelength. This time, use a forward differencing scheme and calculate the derivative $($slope$)$ using the GIVEN x$-$ and y$-$values. The step size will now be set by the x$-$spacing of the model spectrum. Make a plot of the model solar spectrum flux, and do what you did before: color the points where the slope is positive as one color and those where the slope is negative as another color $($you will want to use plt$.$scatter this time$).$ Set the x$-$limits of the plot to zoom into a $50$nm chunk of the spectrum $($any chunk will do$)$ so you can see what is happening.

$8.$ Discuss why the method we used to find the peak of the Planck function would not work to find the peak of the actual Sun$-$like spectrum.

$9.$ How might you fix the problem you identified in the previous question? $($You don$$t need to show the code you would write but rather, speak in pseudocode or generalities about what you might do to overcome the issue.$)$