Let\'s model the Sun\'s energy generation if it was entirely from gravitational

Question

Let's model the Sun's energy generation if it was entirely from gravitational contraction (as hypothesized before we learned about nuclear fusion in stars). A. Write an expression for the potential energy (dU) of a spherical shell of mass dim, in terms of its distance of r from the center, and the mass M_r contained within the radius? B. Assuming a constant density throughout the star (not a good assumption, but let's go with it anyway), write an expression for dm in terms of r, dr, and rho_avg. c. Using the above two, integrate dU with respect to dr to get a total potential energy of the Sun as a result of its collapse from a cloud of nearly infinite size in comparison. D. Knowing the mass and radius of the sun, what is this total energy store in Joules? E. Using solar luminosity of L = 3.839 times 10^26 W, how long could the Sun shine using energy from gravitational contraction alone?

Explanation / Answer

Ans:- (a) potential energy is denoted as (du), a sphere shell of mass dm distance from center to circumference r .

So now Volume = 4 r2 dr

now expression of potential energy of sphere shell

du = - G M (r) dm / r

(b). Constant energy though the star

Density = (r)

Mass = 4 r2 (r) dr

Now Expression on the density constant M(r) = 4/3 r3

(C) by using total potential energy of sun

dU = - (G(4/3 r3 ) (4 r2 dr)) / r

      = - (16 G2-2 / 3 ) r4 dr

integrate du with respect to dr the overall radii . so we find the gravitational potential energy of the full sphere

U = - 16 G2-2 / 3 ) r4 dr

     = - (16 G2-2 / 3) R5 / 5

we know      = M / 4/33

so          U = -3/5 (GM2 /R)

(d) the total energy store the virial theorem

E = 1/2 U

in Initially the gravitational potential energy is littilebit less. so we are fully change in gravitational energy is

U = U initial - U final

         = 0 - (-3 / 5( GM2 / R))

       = 3 / 5( GM2 / R)

So that 1/2 energy can be radiated as the sun compress.

Erad = 3/10 GM2 /R = 1041 Joules

(e). as given the sustain the solar luminosity = 3.893 X 1026 by using Kelvin-Helmholtz timescale formula
      denoted as tKH = E rad / Lsun

                                            = 10 41 / 3.893 X 10 26

                                            = 2.5 X 1014 Sec means Approx 8 X 106 years

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