The initial stage of star formation is the collapse of a gravitationally unstable volume of gaswithin a molecular cloud.a Find the Jeans mass and radius for a molecular cloud at a temperature T = 10 K andnumber density n =106 cm-3. Assume the mean particle mass is umH, where u2.8.Give your answers in solar masses and au, respectively.b Assuming the molecular gas follows the ideal gas law, calculate |dP/dr| ~ JAP/Ar|Pe/ RJ (units [N m-³]) at the beginning of the collapse, where Pe is an approximatevalue for the central pressure of the cloud. Assume that Pmolecular cloud, and the mass and radius are the values determined above.0 at the edge of thec Show that |dP/dr| is significantly smaller than GM,p/r. What can you interpretabout the core's dynamics from this result?d Show that if the excess energy due to the change in gravitational potential energyduring the core's collapse is radiated away efficiently, and the collapse is thereforeisothermal, the contribution of dP/dr in C&O Equation 10.5 continues to decreaserelative to GM,P/r², meaning that dP/dr can be ignored once free-fall collapse begins.(Note that this only applies while the gas is isothermal!)Hint: examine the dependence of dP/dt, given the simplifying assumptions in a), andGM,p/r² on r.

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The initial stage of star formation is the collapse of a gravitationally unstable volume of gas within a molecular cloud. a Find the Jeans mass and radius for a molecular cloud at a temperature T = 10 K and number density n = Give your answers in solar masses and au, respectively. 106 cm-3. Assume the mean particle mass is uMH, where = 2.8. b Assuming the molecular gas follows the ideal gas law, calculate |dP/dr| Pe/ RJ (units [N m-³]) at the beginning of the collapse, where P. is an approximate value for the central pressure of the cloud. Assume that P = 0 at the edge of the molecular cloud, and the mass and radius are the values determined above. JAP/Ar| ~

The initial stage of star formation is the collapse of a gravitationally unstable volume of gas within a molecular cloud. a Find the Jeans mass and radius for a molecular cloud at a temperature T = 10 K and number density n = Give your answers in solar masses and au, respectively. 106 cm-3. Assume the mean particle mass is uMH, where = 2.8. b Assuming the molecular gas follows the ideal gas law, calculate |dP/dr| Pe/ RJ (units [N m-³]) at the beginning of the collapse, where P. is an approximate value for the central pressure of the cloud. Assume that P = 0 at the edge of the molecular cloud, and the mass and radius are the values determined above. JAP/Ar| ~

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter19: Temperature, Thermal Expansion And Gas Laws

Section: Chapter Questions

Problem 43PQ

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The initial stage of star formation is the collapse of a gravitationally unstable volume of gas within a molecular cloud. a Find the Jeans mass and radius for a molecular cloud at a temperature T = 10 K and number density n = 106 cm-3. Assume the mean particle mass is umH, where u = 2.8. Give your answers in solar masses and au, respectively. b Assuming the molecular gas follows the ideal gas law, calculate |dP/dr| = JAP/Ar| ~ Pe/RJ (units [N m-³]) at the beginning of the collapse, where P. is an approximate value for the central pressure of the cloud. Assume that P molecular cloud, and the mass and radius are the values determined above. 0 at the edge of the c Show that |dP/dr| is significantly smaller than GM,p/r². What can you interpret about the core's dynamics from this result?
The following quotation is taken from the article “Quantum Black Holes”, by Bernard J. Carr and Steven B. Giddings, in the May 2005 issue of Scientific American. "The total time for a black hole to evaporate away is proportional to the cube of its initial mass. For a solar-mass hole, the lifetime is an unobservably long 1064 years." a. Recall that the solar mass is 2 × 10³0 kilograms. Write a formula for the lifetime, L, of a black hole as a function of its mass, m. Start by finding the value of the constant k, then write your function using the letter k (rather than its value in scientific notation). For example, for a direct variation you would write “L(m) = km”. k = a × 10¹ where a = L(m) = b. The present age mass = c × 10ª kg, where c = A and b = = of the universe is about 10¹0 years. What would be the mass of a black hole as old as the universe? ID and d = J
a Given a star formation rate of 50 M yr−1, what is the measured Hα flux, in erg/sec? b What types of stars are responsible for this flux? c How do they produce the flux? d Why are they an indicator of star formation?
Models of the first star-forming clouds indicate that they had a temperature of roughly 150 K and a particle density of roughly 400,000 particles per cubic centimeter at the time they started trapping their internal thermal energy. ▼ Part A Estimate the mass at which thermal pressure balances gravity for these values of pressure and temperature. Express your answer in kilograms. —| ΑΣΦ Mcloud Submit Part B = Mcloud How does that mass compare with the Sun's mass? Express your answer in solar masses. Submit Request Answer = ΤΙ ΑΣΦ Request Answer ? ? kg MSun Review
1. a) Using the virial mass for a spherical distribution of stars of radius R and mass M, find how R depends on o and Lassuming some constant M/L. b) Now, assume that all ellipticals have some constant surface brightness to show that L < 04
c) Derive the Schwarzschild criterion for the onset of convection in an ideal gas, namely d ln T d ln P 7-1 Y Explain all steps in your derivation, and justify any assumptions that you make. d) In a region of convective instability near the surface of a solar-type star of total mass M, the temperature and pressure are related approximately by the expression P KT5/2. Show that the temperature gradient for an ideal gas in hydrostatic = equilibrium in this convection zone is given by dT dr 2Gm(r)μ 5Rr² Further, assuming that the mass in the convection zone is small compared to M, show that at a depth h measured from the top of the convection zone, the temperature is approximately given by T = Ts + 2GMμ -h₂ 5RR² when his small compared to R, and Ts is the temperature at the top of the convection zone.
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e https://session.masteringphysics.com/m.. Mas A MyLab and Mastering Canvas Login - Santa Monica College CHomework 8 Exercise 10.43 - Enhanced - with Solution Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly 1014 times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 9.0x105 km (comparable to our sun); its final radius is 18 km. Part A If the original star rotated once in 29 days, find the angular speed of the neutron star. Express your answer in radians per second. For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Anyone can be a ballerina. ? rad/s Submit Request Answer Provide Feedback MacBook Air esc 80 888 F1 F2 F5 F6 F7 F9 @ 2$ & * 2 3 4 6. 8 C Q W R Y S G K C V
(a) In the deep space between galaxies, me density of atoms is as low as 106atoms/m3, and me temperature is a frigid 2.7 K. What is me pressure? (b) What volume (in m3) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?
1. A simple model of a star of radius R assumes (not very realistically!) that the density p is constant. We further assume that the star is made up of pure hydrogen, obeying the ideal gas law. (a) We assume that the gas pressure P drops to zero at the star's surface, e.g., we adopt the boundary condition P(R) = 0; for this part, solve the equations of stellar structure to get the pressure profile P = P(r). (Note: in this particular case, it does not matter whether you use the boundary condition P(R) = 0 or dP/dr = 0 at r=0!) (b) Find the temperature profile T = T(r). (c) If the nuclear energy generation rate ɛ scales with temperature as ɛ * T, determine the radius at which ɛ drops to 10% of its central value. (d) Suppose that ɛ ~ T17. immediately above? What would be the corresponding radius, as computed in (c)
Problem 2: Black hole – the ultimate blackbody A black hole emits blackbody radiation called Hawking radiation. A black hole with mass M has a total energy of Mc², a surface area of 167G²M² /c*, and a temperature of hc³/167²KGM. a) Estimate the typical wavelength of the Hawking radiation emitted by a 1 solar mass black hole (2 × 103ºkg). Compare your answer to the size of the black hole. b) Calculate the total power radiated by a one-solar mass black hole. c) Imagine a black hole in empty space, where it emits radiation but absorbs nothing. As it loses energy, its mass must decrease; one could say "evaporates". Derive a differential equation for the mass as a function of time, and solve to obtain an expression for the lifetime of a black hole in terms of its mass.
what is the answer for sub-item (b) if the radius of the neutron star is 6.676 km? (express your answer in the proper SI unit and without scientific notation) (b)What is the average density of a neuron star that has the same mass as the sun but a radius of only 20.0 km?