Problem 1. Interstellar Cloud Collapse: Jeans Criteria

James Jeans discovered that only clouds of mass greater than a certain threshold

can collapse to form stars. You will reconstruct a derivation of the Jeans Mass.

a. Gravitational Equilibrium: A stable orbit of an object of mass m around another object of mass M requires that the energy of motion or “kinetic” energy of the object be a certain proportion of its gravitational binding energy or “potential” energy. The kinetic energy, KE, is defined

KE = 1/2mv^2

where m is the mass of the object and v is its speed. The gravitational potential energy is defined as

U = -(GMm)/(r)

where G is the gravitational constant of Newton’s so-called Universal Law of Gravitation, M and m are the masses of the interacting objects and r is the radius of the orbit. The total energy, E, of the object of mass m is the sum of its kinetic and potential energies, E = KE + U. Note the minus sign in the definition of potential energy. This can be interpreted as the amount of energy an object has to overcome in order to escape the gravitational influence of another object. (Imagine rolling a marble up the surface of the inside of a bowl - see below. The well of the bowl represents the negative gravitational binding or potential energy. If the marble is not given sufficient kinetic energy, it remains inside the bowl - i.e. bound to the gravitational potential).

Use the Universal Law of Gravitation and Newton’s Second Law of motion, along with the aforementioned definitions of kinetic and potential energy to develop the algebraic relationship between kinetic and potential energy for a circular orbit (this is also known as the Virial Theorem).

b. Kinetic Energy and Temperature: We will construct the algebraic relationship between kinetic energy and temperature for a gas from the ideal gas law, PV=NkT where P = pressure, V = volume, N = number of particles, k is Boltzmann’s constant, and T is temperature. Recalling that pressure can be expressed as P=F/A,where F = force and A = area, rewrite the ideal gas law in terms of Newton’s Second Law of motion in order to relate kinetic energy to temperature using dimensional analysis. (HINT: keep in mind the definitions of the kinematic variables

ai = vi/t, i = x or y or z

and .

Vx= x/t, Vy = Y/t, Vz = Z/t

Assume there is temperature associated with the kinetic energy in each direction [x,y,z] of motion

in 3-D [3 Dimensional] space.)



c. The Jeans Mass:

i. (NOTE: Treat the cloud as two equal masses interacting gravitationally across a distance equal to the radius of the cloud.) Use your result from part a to write down the condition for gravitational collapse in terms of the kinetic and potential energies (NOTE: This condition is an INEQUALITY)



ii. Use your result from part b in order to replace the kinetic energy with its temperature equivalent in your expression for the collapse condition.



iii. Solve the expression in ii above for the mass.

Question

Problem 1. Interstellar Cloud Collapse: Jeans Criteria

James Jeans discovered that only clouds of mass greater than a certain threshold

can collapse to form stars. You will reconstruct a derivation of the Jeans Mass.

a. Gravitational Equilibrium: A stable orbit of an object of mass m around another object of mass M requires that the energy of motion or “kinetic” energy of the object be a certain proportion of its gravitational binding energy or “potential” energy. The kinetic energy, KE, is defined

KE = 1/2mv^2

where m is the mass of the object and v is its speed. The gravitational potential energy is defined as

U = -(GMm)/(r)

where G is the gravitational constant of Newton’s so-called Universal Law of Gravitation, M and m are the masses of the interacting objects and r is the radius of the orbit. The total energy, E, of the object of mass m is the sum of its kinetic and potential energies, E = KE + U. Note the minus sign in the definition of potential energy. This can be interpreted as the amount of energy an object has to overcome in order to escape the gravitational influence of another object. (Imagine rolling a marble up the surface of the inside of a bowl - see below. The well of the bowl represents the negative gravitational binding or potential energy. If the marble is not given sufficient kinetic energy, it remains inside the bowl - i.e. bound to the gravitational potential).

Use the Universal Law of Gravitation and Newton’s Second Law of motion, along with the aforementioned definitions of kinetic and potential energy to develop the algebraic relationship between kinetic and potential energy for a circular orbit (this is also known as the Virial Theorem).

b. Kinetic Energy and Temperature: We will construct the algebraic relationship between kinetic energy and temperature for a gas from the ideal gas law, PV=NkT where P = pressure, V = volume, N = number of particles, k is Boltzmann’s constant, and T is temperature. Recalling that pressure can be expressed as P=F/A,where F = force and A = area, rewrite the ideal gas law in terms of Newton’s Second Law of motion in order to relate kinetic energy to temperature using dimensional analysis. (HINT: keep in mind the definitions of the kinematic variables

ai = vi/t, i = x or y or z

and .

Vx= x/t, Vy = Y/t, Vz = Z/t

Assume there is temperature associated with the kinetic energy in each direction [x,y,z] of motion

in 3-D [3 Dimensional] space.)

c. The Jeans Mass:

i. (NOTE: Treat the cloud as two equal masses interacting gravitationally across a distance equal to the radius of the cloud.) Use your result from part a to write down the condition for gravitational collapse in terms of the kinetic and potential energies (NOTE: This condition is an INEQUALITY)

ii. Use your result from part b in order to replace the kinetic energy with its temperature equivalent in your expression for the collapse condition.

iii. Solve the expression in ii above for the mass.

E>0, KE> U, object
escapes potential well.
E=0, KE = U, object
just reaches edge of
well.
E =0
E< 0, KE< |U], object
remains in potential
well.
E=-JU|, KE=0, object
remains at bottom of
well Total energy is at
min of potential
E =-|U|

Accepted Answer

a. For a circular orbit,

E>0, KE> U, object escapes potential well. E=0, KE = U, object just reaches edge of well. E =0 E< 0, KE< |U], object remains in potential well. E=-JU|, KE=0, object remains at bottom of well Total energy is at min of potential E =-|U|