University:
Massachusetts Institute of Technology
Course:
8.282J | Astrophysics
Academic year:

2023
Views:

114

Pages:

2
Author:

Vance Pace

Earth, Atmospheric, and Planetary Sciences Department Computational Problem Set 2 Polytropic Models for Stars Optional 1. Integrating the LaneEmden Equation Integrate the LaneEmden equation: 1 d ξ 2 dξ � ξ 2 dφ � dξ = −φn for polytropic indices of n = 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5. Break up this second order diﬀerential equation into two ﬁrstorder, coupled equations in dφ/dξ ≡ u and du/dξ. Use a 4thorder RungeKutta integration scheme or some other equiva lent integration method to ﬁnd φ(ξ). Recall the boundary conditions at the center: u(0) = 0 and φ(0) = 1 . Use the analytic expansion for φ(ξ) near the center: ξ2 6 to help start the integration. The surface of the star is deﬁned by φ(ξ1 ) = 0. φ(ξ) = 1 − Plot the dimensionless temperature, φ(ξ), and the dimensionless density, φn (ξ), for all 6 values of n. It would be best to put all the temperature plots on one graph and all the density plots on another. 2. Tabulating Some Physical Properties of Polytropes As the integrations in part (1) are underway, compute for each model the dimensionless potential energy, Ω (in units of −GM 2 /R), and the dimensionless moment of inertia, k (in units of M R2 ). Tabulate ξ1 , −(dφ/dξ)ξ1 , Ω, and k for each of the 6 polytropic models. 3. Model of the Sun Use an n = 3 polytropic model to represent the internal structure of the Sun. The two param eters to ﬁx are M = M� and R = R� . (a) Plot the physical temperature (in K) vs. radial distance in units of r/R� . Plot log T vs. r/R� . Do the same for the physical density (g cm−3 ). Again, plot log ρ vs. r/R� . Instead of using the values for the central density, ρ0 , and central temperature, T0 , deduced for an n = 3 polytrope with M = M� and R = R� , take the “known” values of ρ0 = 158 g cm−3 and T0 = 15.7 × 106 K. 1 (b) Compute the nuclear luminosity of the sun using the above temperature and density proﬁles. Take the nuclear energy generation rate to be: −2/3 �(ρ, T ) = 2.46 × 106 ρ2 X 2 T6 −1/3 exp(−33.81T6 ) ergs cm−3 sec−1 where ρ is in g cm−3 , T6 is the temperature in units of 106 K, and X is the hydrogen mass fraction (take X = 0.6). Note that the units of �(ρ, T ) are ergs cm−3 sec−1 and not ergs g−1 sec−1 as given in the lecture. Reduce the problem to a dimensioned constant times an integral involving only functions of φ and ξ. (There will also appear a “T0 ” inside the integral for which you can plug in the value of 15.7 × 106 K, or T06 = 15.7.) Show the value of your constant and the form of the dimensionless integral. Evaluate the nuclear luminosity of the Sun in units of ergs sec−1 . 2