Question

could you please solve a and b?

Chapier 2i. Note: you needn't derive Kepler's laws-but do mention when you are using them, an describe the physical concepts involved and the meanings behind the variables. u) Consider two stars Mi and M; bound together by their mutual gravitational force (and isolated from other forces) moving in elliptical orbits (of eccentricity e and semi-major axes ai and az) at distances 11 in n and r from their center of mass located at the common focus of the two ellipses (see Figure 2. the text). (a) Starting with the equation of motion r(0) for a closed (bound) orbit, demonstrate the relationships between the periastron (or perihelion, perigee, etc.) and apastron (or aphelion, apogee, etc.) distances, re and rA, in terms of the system semi-major axis a and the orbit eccentricity s (b) Write down an expression for the sum of the kinetic and gravitational potential energies, involving each of the masses Mi and M,, their separation r r -r + ri and their orbit speeds vi and This sum is time-invariant (conserved) and called the total orbital energy E. Now use the relationship between vi and v and system speed v (v Vi +v2), the definition of reduced mass pu, and the conservation of system's linear momentum in the absence of external forces (Mivi M2v2 Hv) to express t conservation of energy relation as: In these coordinates, the reduced mass orbits the center of mass that contains the sum of the masse that is located at one of the foci of the system ellipse of the same eccentricity e as that of each of tl separate orbits, and with a semi-major axis a = al + a2-The same radial vector r-n-n now connects M to μ, and μ moves along this ellipse with the system velocity v = v-vi (see Figure 2.12 in the text) (c) Apply the Virial Theorem (Eq. 2.46) to find that EGM/2a for bound (and so periodic) orbits. You may use the fact that <1/ I/a, where<>represents a time-average quantity over the (extra credit 10 pts.: derive this relation yourself-i.e, do Problem 2.9). Now, solve the orbital energy equation for the system orbital speed v; it should involve M, r, a, and constants. pe (d) Beginning with your expression of v, and using the expression for the conservation of linear momentum (Miv.-My,-μν) in the absence of external forces, determine the orbital speeds (v1. v) of each of the two masses (Mi, M:). Express them in terms of M, r, a, Mi and M:, and constants (e) Rewrite these expressions for vi and v in the limit that Mi >>M2 (as for a planet with mass M orbiting a star of mass Mi); but do not assume Mi is infinite or M2-0. Next, assume the Sun to be M and Earth to be M2 and compute Earth's "typical" orbital speed v(r- a), expressing it in km/s (0) Now set r equal to rp and rA in your full expression for v to derive the system speeds at these two turn- around points of the elliptical orbit, ve and va, as given in Equations 2.33 and 2.34 on p.47 of your textbook. Then write down a simple expression for their ratio, ve/vA, involving the eccentricity s (2) Finally, starting with the definition L-rxp, write down expressions for the system orbital angular momentum L at the two turn-around points of the orbit (aphelion, perihelion). Remember: at these t points v ve (here, v,- 0!), so while L-Li+L2 is a constant of motion (Li and L are each also constants of motion) in the absence of external forces, it is easiest to compute at the turn-around points Substitute your expressions for ve and ry into the expression for the system orbital angular momentum L as calculated at perihelion (or vA and ra into the expression for L at aphelion) to determine an expression for the system angular momentum in terms of variables μ M, a, e, as in Equation 2.30. Provide a brief conceptual explanation for the behavior of L with e (for fixed values in p, M, a). over) Congratulations

Answer 1

(a) For a closed bound orbit, Let r (theta) = 1/u (theta) Using inverse square law formula i.e., d^2 u/d theta^2 + u = m k/L^2 (i) Where, f (r) = -k/r^2 Vertical-bar L vector Vertical-bar = m r^2 theta^. We can deduce, l/r = 1 + epsilon cos (theta - theta_0), where, l = L^2/m k: epsilon = squareroot 1 + 2 E l/k Here, m rightarrow mu (reduced mass) \r\n Now, perihelion and aphelion position of r-motion are turning points given by theta = theta_0 and theta = theta_0 + pi So, at perihelion position. r = r_p gives l/r_p = 1 + epsilon at aphelion position: - r = r_A l/r_A = 1 - epsilon now, semi-major axis a = r_A + r_r/2 = l/2 (1/1 + epsilon + 1/1 - epsilon) = l/1 + epsilon^2 a = -k/2 E (E < 0) So, r_p = l/1 + epsilon = a(1 - epsilon^2)/1 + epsilon