EP3 Neutron stars are often formed in supernova explosions. They are extremely d

Question

EP3 Neutron stars are often formed in supernova explosions. They are extremely dense stars comprised primarily of neutrons, (a) For a certain density range (around 1012 g/cm3) the pressure P within the star is given by an equation of the form P = kh2Mm pn, where A: is a dimensionless constant, h is Planck's constant (with the dimensions of energy times time), M is the mass of a single neutron, and p is the mass density (mass/volume) of the star. Find the exponents m and n using dimensional reasoning, (b) Find an expression for the speed of sound waves within the star, in terms of h, M, k, and p. (c) At still higher densities (of order 1016 g/cm3) the pressure within the star is proportional to the four-thirds power of density (P ~ p4/3), and P depends also upon h, M, and c, the speed of light. How does P depend upon these three quantities? (d) Find an expression for the speed of sound within the star in this density range, in terms of h, M, c, p, and a dimensionless constant.

Explanation / Answer

(a)

M - Mass , L - Length T - Time

Pressure = Force / Area = MLT-2 / L-2

Pressure = ML-1T-2

h (planks' constant) = E / f = F * s / f = MLT-2 * L / T-1

h = ML2T-1

Rho(Mass density) = mass / volume = ML-3

P = kh2Mm(rho)n

ML-1T-2 = (ML2T-1)2 Mm (ML-3)n

ML-1T-2 = M(2+m+3n) L(4-3n) T-2

Comparing LHS and RHS

2+m+3n = 1, 4 - 3n = -1

n = 5/3, m = -6

(b)

Speed of sound waves = LT-1

Speed = k hh Mm (rho)n

LT-1 = (ML2T-1)h Mm (ML-3)n

Comparing coefficients, we get :

For M: 0 = h + m + n

For L: 1 = 2h - 3n

For T: -1 = -h - 3n

After solving:

h = 2/3

n = 1/9

m = -7/9

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(c)

P = k (rho)4/3 hh Mm cc

ML-1T-2 = (ML-3)4/3 (ML2T-1)h Mm (LT-1)c

Comparing coefficients, we get :

For M: 1 = 4/3 + h + m

For L: -1 = -4 + 2h + c

For T: -2 = -h - c

After solving:

h = 1, c = 1, m = -4/3

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(d)

Speed = k (rho)r hh Mm cc

LT-1 = (ML-3)r (ML2T-1)h Mm (LT-1)c

Comparing coefficients, we get :

For M: 0 = r + h + m

For L: 1 = -3r + 2h + c

For T: -1 = -h - c

This has 4 variables therefore infinitely many answers are possible.

Assuming say r = 4/3 (speed also depends on (rho)4/3 )

After solving:

h = 4, c = -3, m = -16/3

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