Problem 4 In 1933 Fritz Zwiky used the virial theorem to demonstrate the existence of dark...
Problem 4
In 1933 Fritz Zwiky used the virial theorem to demonstrate the existence of dark matter (even though nobody believed him for the subsequent 40 years) in the Coma cluster of galaxies. The demonstration assumes that the Coma galaxies have a random spherical distribution around the center of gravity of the cluster, and that their velocity is also distributed randomly. With these hypothesis the average potential energy of the cluster can be approximated with the potential energy of a spherical mass M of radius R: <U> ≈ −3/5 * GM^2/R and the average kinetic energy of the cluster (assuming that the average velocity of the galaxies in the cluster is zero) is: <K> ≈ 1/2 * M(3σr^2) where σr is the dispersion in the radial velocities of the galaxies in the cluster (the factor 3 in the formula derives from the assumption that the velocity is equally distributed in the three spatial components — radial and the two orthogonal components).
a. Use the virial theorem and the two equations above to derive the expression of the mass M for the cluster (a mass derived in this way is called “virial mass”).
b. Zwiky measured a radial velocity dispersion σr ' 977 km/s and a radius R ' 3 Mpc (= 3 × 106 pc). What is the virial mass of the cluster, in unit of solar masses (1 M ' 2 × 1030 kg)?
c. What is the mass-to-light ratio of the cluster (the ratio of the mass in solar masses and its luminosity in solar luminosities), if the measured luminosity of the cluster is L = 5 × 1012 L?
d. Can you guess why this result lead Zwiky to suggest the existence of undetected “dark matter”? Hint: if all mass in the galaxies was in the form of luminous stars, the mass-to-light ratio would be close to the ratio of the mass of the average star with the luminosity of the average star, and the Sun is quite close to the average star in a galaxy.
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Problem 4
In 1933 Fritz Zwiky used the virial theorem to demonstrate the
existence of dark...
We used velocity dispersion and average distance between galaxies in the cluster to determine the virial mass, and we used the number of galaxies and the average mass of a galaxy to determine luminous...
We used velocity dispersion and average distance between galaxies in the cluster to determine the virial mass, and we used the number of galaxies and the average mass of a galaxy to determine luminous mass. How would the average distance between galaxies have to change in order to eliminate the evidence for dark matter in galaxy clusters? Calculate the change needed in the case of the Coma Cluster.
1) The missing mass problem is a) starlight is observed, but there is no evidence of...
1) The missing mass problem is a) starlight is observed, but there is no evidence of mass b) the amount of mass measured is less than the amount of light measured c) starlight follows the same distribution as mass d) there are indications of mass from motion but no light e) spacetime is warped by mass 2) The rotation curve of the Galaxy is a) flat in the core, planet-like in the disk b) wheel-like in the core, flat in...
cosmology (1) 1. (10 points] In a simple model for a spherically symmetric dark matter halo,...
cosmology
(1) 1. (10 points] In a simple model for a spherically symmetric dark matter halo, the dark matter density is given by M PDM = 4ar, where M, is the enclosed mass at the virial radius r. Let v(r) be the velocity of a dark matter particle that is executing a circular orbit in the halo with radius r centered at r=0. Show that v(r) is independent of radius and obtain its value in terms of M, and ty....
LP 5. (20 pts) It is known that most of the mass of galaxies is some exotic material that doesn't interact with light or normal matter. Let's assume that this "dark matter" is due...
LP 5. (20 pts) It is known that most of the mass of galaxies is some exotic material that doesn't interact with light or normal matter. Let's assume that this "dark matter" is due to a new elementary particle---alaskonium. Like Alaska in the winter, alaskonium in the galaxy is cold and dark. We will assume that alaskonium is a fermion and that it is so cold that it is degenerate; a gravitationally bound, nonrelativistic, degenerate gas. Take the total mass...
2. Mass versus Light One of the primary goals of every galaxy survey is to measure the distributi...
2. Mass versus Light One of the primary goals of every galaxy survey is to measure the distribution of the luminosities of galaxies, which has been found to be well parameterized by the luminosity function L/L- where φ(L)dL is the number density of galaxies with luminosity between L and L + dL, and φ*, L*, and a are parameters to be determined from galaxy surveys (a) Show that the luminosity density of galaxies, J, is related to ф*, L*, and...
2. Mass versus Light One of the primary goals of every galaxy survey is to measure the distributi...
I want an answer before Monday.
2. Mass versus Light One of the primary goals of every galaxy survey is to measure the distribution of the luminosities of galaxies, which has been found to be well parameterized by the luminosity function L/L- where φ(L)dL is the number density of galaxies with luminosity between L and L + dL, and φ*, L*, and a are parameters to be determined from galaxy surveys (a) Show that the luminosity density of galaxies, J,...
Question 1 (10 marks) (a) (4 marks) Recall that the gravitational potential energy for two masses...
Question 1 (10 marks) (a) (4 marks) Recall that the gravitational potential energy for two masses is Ug-GMm Use this fact to show that the virial theorem holds for a mass m, executing uniform circular motion about a much larger mass M (b) (3 marks) The potential energy of a collection of N particles each of mass m in a region with radius R can be written as Use this expression to derive a mass estimate for the virialized system...
Please use the formulate sheet and show all steps to make sure the question is worth...
Please use the formulate sheet and show all steps to make sure
the question is worth any points
a) The initial ratio of deuterium (D) to hydrogen (H) in a planet's atmosphere was 175000; however, the present ratio is 1/1500 and the initial and final abundances of D are 5 x 10° atoms per m3 and 9 x 106 atoms per m2, respectively. What fraction of deuterium has been lost, and what fraction of hydrogen has been lost in this...
could you please solve a and b? Chapier 2i. Note: you needn't derive Kepler's laws-but do...
could you please solve a and b?
Chapier 2i. Note: you needn't derive Kepler's laws-but do mention when you are using them, an describe the physical concepts involved and the meanings behind the variables. u) Consider two stars Mi and M; bound together by their mutual gravitational force (and isolated from other forces) moving in elliptical orbits (of eccentricity e and semi-major axes ai and az) at distances 11 in n and r from their center of mass located at...
Problem 4: Read Appendix 2 below (Sec. 1.4.1 of Kasap) and then solve. A metallic back...
Problem 4: Read Appendix 2 below (Sec. 1.4.1 of Kasap) and then solve. A metallic back contact is applied to the CdTe solar cell of Problem 1 using a set up similar to that described in Figure 1.74 (b) on the next page. To form the metallic back contact, two evaporation sources are used, Cu and Au. An initial 3 nm layer of Cu is deposited first and then 30 nm of Au is deposited. After these depositions, the sample...
cosmology
(1) 1. (10 points] In a simple model for a spherically symmetric dark matter halo, the dark matter density is given by M PDM = 4ar, where M, is the enclosed mass at the virial radius r. Let v(r) be the velocity of a dark matter particle that is executing a circular orbit in the halo with radius r centered at r=0. Show that v(r) is independent of radius and obtain its value in terms of M, and ty....
LP 5. (20 pts) It is known that most of the mass of galaxies is some exotic material that doesn't interact with light or normal matter. Let's assume that this "dark matter" is due to a new elementary particle---alaskonium. Like Alaska in the winter, alaskonium in the galaxy is cold and dark. We will assume that alaskonium is a fermion and that it is so cold that it is degenerate; a gravitationally bound, nonrelativistic, degenerate gas. Take the total mass...
2. Mass versus Light One of the primary goals of every galaxy survey is to measure the distribution of the luminosities of galaxies, which has been found to be well parameterized by the luminosity function L/L- where φ(L)dL is the number density of galaxies with luminosity between L and L + dL, and φ*, L*, and a are parameters to be determined from galaxy surveys (a) Show that the luminosity density of galaxies, J, is related to ф*, L*, and...
I want an answer before Monday.
2. Mass versus Light One of the primary goals of every galaxy survey is to measure the distribution of the luminosities of galaxies, which has been found to be well parameterized by the luminosity function L/L- where φ(L)dL is the number density of galaxies with luminosity between L and L + dL, and φ*, L*, and a are parameters to be determined from galaxy surveys (a) Show that the luminosity density of galaxies, J,...
Question 1 (10 marks) (a) (4 marks) Recall that the gravitational potential energy for two masses is Ug-GMm Use this fact to show that the virial theorem holds for a mass m, executing uniform circular motion about a much larger mass M (b) (3 marks) The potential energy of a collection of N particles each of mass m in a region with radius R can be written as Use this expression to derive a mass estimate for the virialized system...
Please use the formulate sheet and show all steps to make sure
the question is worth any points
a) The initial ratio of deuterium (D) to hydrogen (H) in a planet's atmosphere was 175000; however, the present ratio is 1/1500 and the initial and final abundances of D are 5 x 10° atoms per m3 and 9 x 106 atoms per m2, respectively. What fraction of deuterium has been lost, and what fraction of hydrogen has been lost in this...
could you please solve a and b?
Chapier 2i. Note: you needn't derive Kepler's laws-but do mention when you are using them, an describe the physical concepts involved and the meanings behind the variables. u) Consider two stars Mi and M; bound together by their mutual gravitational force (and isolated from other forces) moving in elliptical orbits (of eccentricity e and semi-major axes ai and az) at distances 11 in n and r from their center of mass located at...
Problem 4: Read Appendix 2 below (Sec. 1.4.1 of Kasap) and then solve. A metallic back contact is applied to the CdTe solar cell of Problem 1 using a set up similar to that described in Figure 1.74 (b) on the next page. To form the metallic back contact, two evaporation sources are used, Cu and Au. An initial 3 nm layer of Cu is deposited first and then 30 nm of Au is deposited. After these depositions, the sample...
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