ps1_question4

ps1_question4 January 24, 2024 1 Question 4 I’m here to guide you through making your own hubble diagram! First, lets load in some code packages to make our lives easier: [31]: import matplotlib.pyplot as plt from scipy.optimize import curve_fit import numpy as np import pandas as pd Now,lets load in our data: [32]: cepheid_periods = np . loadtxt( "cepheid_periods.txt" ) cepheid_magnitudes = np . loadtxt( 'cepheid_magnitudes.txt' ) wavelengths = np . loadtxt( 'wavelengths.txt' ) 1.1 Part a) Now that you’ve gotten to do some analysis of cepheid variables, lets make our own hubble diagram with some real data! We’ve provided a python notebook to help you complete this question, 1.1.1 Finding Distance We have observed cepheid variables in 10 assorted host galaxies . In table 1, we give the periods (T) and apparent magnitudes (m) of these cepheids. Find the distances between our observer and each host galaxy by filling in your formula for distance. Note: Assume cepheids have an approximate period luminosity relationship of: 𝑀 = −3(𝑙𝑜𝑔 10 (𝑃𝑒𝑟𝑖𝑜𝑑) − 1) − 4 Where period is given in units of days. [33]: ## Define a function to calculate cepheid distance from aparent mangitude and ␣ ↪ period def cepheid_distance_finder (magnitude,period): return 10 ** ((magnitude - ( -3 * (np . log10(period))) + 4 ) / 5 ) Now we just run this function on our lists of periods and aparent magnitudes to get distances 1

[34]: # plug in your data to find the cepheid distances cepheid_distances = cepheid_distance_finder(magnitude = cepheid_magnitudes, ␣ ↪ period = cepheid_periods) print (cepheid_distances) #just to double check my function is right [ 1767591.89944544 15002392.53348405 1559519.7631191 814398.39674279 9974491.17060978 4296137.67397539 3759161.73526182 1262249.66649162 6657474.48808416 1696805.72732787] 1.2 Part b) 21 cm emission lines are easily recognizable emission lines that have been detected from all of these listed galaxies. When a 21cm photon would be emitted or absorbed, its wavelength would be about 0.21106114 m (hence 21cm!). In table 2, we have provided the wavelength at which this line was observed on earth. Using this, calculate the recessional velocity of each galaxy [35]: ## Define a function to calculate galaxy recessional velocity using the ␣ ↪ observed wavelength of its 21 cm line c = 3E+5 def velocity_21cm (wavelength): return c * ((wavelength / 0.21106114 ) - 1 ) [36]: ## Plug in your data to find recessional velocity recessional_velocities = velocity_21cm(wavelengths) print (recessional_velocities) [ 547.49064655 1905.59948648 321.29078806 143.98671399 1637.09908892 717.58827797 922.49572802 133.49686257 635.38934737 513.29202524] 1.3 Part c) Now that we have distances and recessional velocities, let’s find Hubble’s Constant! Plot your distances and recessional velocities and fit a line whose slope will determine 𝐻 0 . Please give 𝐻 0 in units of km Mpc −1 s −1 Plot your hubble diagram below: Be mindful of your units! Please include them in your plot [57]: ## Fill out the title and x and y axis labels for your hubble diagram plt . scatter(cepheid_distances, recessional_velocities) # scatter plot of ␣ ↪ recessional velocities for hubble diagram plt . xlabel( 'Distances (Parcsecs)' ) plt . ylabel( 'Recessional Velocities (km/s)' ) plt . title( "Hubble's Law" ) plt . xscale( 'log' ) plt . yscale( 'log' ) plt . xlim( 5e+5 , 5e+7 ) 2