Homework 1 (Due 2/7 at 5pm in Room 1005) 1. a) Consider a spherically symmetric star in hydrostatic equilibrium. The gravitational potential φ satisfies the Poisson equation, ∇2 φ = 4πGρ, (1) where ρ...

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Homework 1 (Due 2/7 at 5pm in Room 1005) 1. a) Consider a spherically symmetric star in hydrostatic equilibrium. The gravitational potential φ satisfies the Poisson equation, ∇2 φ = 4πGρ, (1) where ρ is the density. Assume the density varies as ρ(r) = ρc (1− β r2), (2) where ρc is a constant (the central density). Solve for the pressure as a function of radius. The star is isolated, thus the boundary condition is p(a) = 0, where a is the radius of the star. b) Consider now a star with uniform density ρ0, which has the same mass and radius as the previous star (for appropriate choice of β). Show that if ρ0 is twice the density given by Eq. (2) at r = a, then the central pressure found in a) is a factor of (13/8) larger than that for the uniform star. 2. In a steady 2D flow, the cartesian component trajectories of infinitesimal fluid elements is given by x(t) = r0 cos(ωt), y(t) = r0 sin(ωt), (3) a) From these trajectories determine the Lagrangian velocity components dx/dt and dy/dt and convert these to Eulerian velocity field compo- nents vx(x, y) and vy(x, y). b) Compute the Lagrangian acceleration components d2x/dt2 and d2y/dt2, and show that they agree with the total (or material) derivative ∂/∂t+ v · ∇ of the Eulerian velocity field components determined in a) 3. a) Consider an axisymmetric source-free flow in the (x, y) plane, where the velocity field v = v(r)θ̂ (in polar coordinates). Show that the vorticity is in the z-direction equal to ω = ∂v/∂r + v/r. b) Consider the case of rigid rotation, that is a flow that rotates with angular velocity Ω about the z axis. Show that ω = 2 Ω. c) Consider now another simple 2D flow, that of a shear flow where the only non-vanishing component of the velocity is vx = v0e −y2/L2 . Show that even though all streamlines are parallel to the x-axis, there is non-zero vorticity. Do a sketch of the flow and explain why the sign of the vorticity changes with y.