Problem 1: Consider these data for Cepheids in the galaxy M100, including luminosities derived using Leavitt's Law from observed periods. Cepheid 1: L=3.9 x 103 W. Flux=9.3 x 107' W/m?. Cepheid 2: L=1.2 x 103 W. Flux=3.8 x 1071 W/m?. Cepheid 3: L=2.5 x 10* W. Flux=8.7 x 10~ W /m?, Compute the distance to M100 with data from each of the three Cepheids. Based on your re- sults, estimate the uncertainty in the distance to M100 from this method. (That is, estimate an average value, and an uncertainty in that valueeither compute a standard deviation, or else choose an uncertainty that encompasses the different measurements you computed.) Problem 2: Imagine you've obtained spectra for 3 galaxies and measured the wavelength of the emission line of hydrogen that has a rest wavelength of 656.3 nm. Galaxy 1 shows a wavelength of 659.6 nm, Galaxy 2 has 664.7 nm, and Galaxy 3 has 679.2 nm. a) Calculate the redshift z for each galaxy. b) From their redshifts, calculate the speed each galaxy is moving away from us; give these in km /s, and as a fraction of c. c) Use Hubble's law to estimate the distance to each galaxy. Assume Ho=22 km/s/Mly.Problem 3: When two spiral galaxies collide, the stars very rarely collide, while the gas clouds often do collide, triggering a burst of star formation. a) Estimate the probability that our Sun would collide with another star in the Andromeda Galaxy when the galaxies collide (assume the galaxies collide like two hands clapping, pre- senting the maximum area to each other when they collide). To simplify this, assume each galaxy has 100 billion stars just like the Sun, evenly spread over a flat circular disk with a diameter of 100,000 light years. (Hint; calculate the area of 100 billion circles with the Sun's radius, and compare to the area of the Andromeda galaxy.) b) Estimate the probability that one gas cloud in our Galaxy would collide with another molecular cloud in the Andromeda Galaxy when the galaxies collide. Assume that each 1 galaxy contains 6,000 clouds of warm hydrogen gas, each having a radius of 100 light-years, spread over the same circular disk as above. c) Using the probabilities from a and b above, estimate how many stars will probably collide with other stars; and how many gas clouds will collide with other gas clouds. Describe in a sentence how these explain what we see happening in e.g. the Antennae Galaxies (Fig. 21.5 in textbook). Problem 5: Look at the graph of orbital speed vs. distance for the spiral galaxy NGC 801, Figure 23.4 (or video lecture 13.2). a) Estimate the orbital velocity of stars at a distance of 25,000 light-years from the centre of NGC 801 (roughly the Sun's distance from the centre of our galaxy). Then use the orbital velocity law to determine the mass (in solar masses) within that distance from its centre. b) Estimate the orbital velocity of stars at a distance of 150,000 light-years from the centre of NGC 801. Then use the orbital velocity law to determine the mass (in solar masses) within that distance from its centre. c) Based on your answers to parts a and b, what can you say about the distribution of mass in this galaxy? (Specifically, where is the majority of the mass?) Problem 4: What was the peak wavelength of the cosmic background radiation at the time light left the most distant galaxies we can currently see (at a cosmological redshift of z=10)?7 What is the temperature corresponding to that peak wavelength? p - N - - o -~ Problem 6: A cluster of galaxies has a radius of 5.1 million light-years and its hot gas has a temperature of 6 x 107 K. Estimate the mass of the cluster, in kg and in solar masses. If the combined luminosity of all stars in the cluster is 8 x 10'? Lsun, what is the cluster's mass-to-light ratio