Calculate the change in length of year due to change in Sun's mass

[NewSoul]

EDIT: It seems that just after posting this, my professor decided this problem is too difficult for us to be responsible for. I can't delete this question, so don't feel obligated to answer it. But who knows? Maybe someone else out there has the same question.

Homework Statement

The Sun loses mass at a rate of 3.64E9 kg/s. During the 5,000-yr period of recorded history, by how much has the length of the year changed due to the loss of mass from the Sun? Suggestions: Assume the Earth's orbit is circular. No external torque acts on the Earth–Sun system, so the angular momentum of the Earth is constant.

M₂ = current mass of sun = (Wolfram Alpha is source): 1.988E30 kilograms
r = average distance between sun and Earth (Wolfram Alpha is source): 1.496E11 meters
G = 6.67E-11 m³ kg^-1 s^-2
5000 yr = 1.577E11 s
ΔM = 3.64E9 kg/s

Homework Equations

Kepler's third law: T² = (4π²r³)/GM

The Attempt at a Solution

To solve the equation, I plan on finding the value for T for today (T₂) and the value of T for 5000 years ago (T₁). I did some Googling and someone found the answer to be ΔT = 1.82E-2 s. I don't know if it is correct or not.

Okay, so here's how I started. M₂ (current mass of sun) = 1.988E30 kg. I wanted to find M₁ (previous mass of sun), so I did M₁ = 1.988E30 kg + (3.64E9 kg/s)(1.577E11 s) = 1.988E30 kg.

So, according to my equation above, M₁ and M₂ are almost exactly the same and the difference is negligible. How do I go about this?

Thanks.

 

[SteamKing]

Have you thought about finding the change in period T with respect to the change in mass M from the Kepler relation? You know, derivatives and all that jazz.

 

[rude man]

You need to combine Kepler's 2nd and 3rd laws to get your answer.

The second law reflects the need to conserve angular momentum.
The 3rd law equates gravitational and centripetal forces.

These two equations would enable you to solve for the change in T and also r if desired.

The math is messy but not impossible ...