代做MiA Coursework 3: Dynamics of a rotating star帮做R程序

日期：2024-03-17 12:32

MiA Coursework 3:

Dynamics of a rotating star

Practical instructions: Feel free to submit handwritten solutions for all mathematical derivations (no need to type them up in LATEX, as long as they are readable). Feel free to use any programming language to generate the figure required in Q(4.c) and to compute the numerical values required in Q(4.d). For this assignment, there is no need to upload the source code.

Introduction: Consider a rigidly rotating, self-gravitating, homogeneous fluid system enclosed by an ellipsoidal surface defined as

with Cartesian coordinates x = (x1, x2, x3) and semi-major axes a1 ≥ a2 ≥ a3. The system, being homogeneous, has constant density ρ = M/V , where M is the total mass and the volume is given by

The system, being rigidly rotating, spins with constant angular velocity  and does not display any internal motions. For clarity, all the above quantities (ρ, M, a1, a2, a3, V , Ω) are positive definite. Finally, it is convenient to define the eccentricity and ellipticity of the system as

such that e, η ∈ [0, 1]. Note that, from Eqs. (3)-(4), it follows that  and  Consequently, an oblate ellipsoid is characterised by η = 0 (i.e., a2 = a1) and a prolate ellipsoid is characterised by e = η (i.e., a2 = a3). The spherical limit of the system is recovered by considering e → 0 and η → 0, i.e. a1 = a2 = a3 = a, which is the radius of the corresponding sphere.

1 Inertia moment

Recall (from W4 lectures) that the inertia moment tensor of a self-gravitating fluid system is defined by

Show that, in this case, Iij is diagonal and compute the explicit expressions of the three diagonal terms.

2 Gravitational potential

(a) It is a bit laborious (hence omitted), but it can be proved (see Chandrasekhar 1969, Ellipsoidal Figures of Equilibrium, Chapter 3) that the gravitational potential of the system inside the ellipsoidal surface is given by

where G is the gravitational constant and the coefficients Ai are given by

where

Show that, in the spherical limit (i.e., a1 = a2 = a3 = a), the gravitational potential inside the system introduced at Eq. (6) becomes

for r < a.

(b) It can also be proved (see Chandrasekhar 1969, Ellipsoidal Figures of Equilibrium, Chapter 3) that the three coefficients Ai introduced at Eq. (7) can be explicitly expressed, for a general triaxial ellipsoid, as follows

where

are incomplete elliptic integrals of the first and second kind.

Show that, for an oblate ellipsoid (i.e. with η = 0), A1 = A2 (as expected from the axisymmetry of the oblate case) and compute the explicit expressions of A1, A2, and A3 as functions of e.

Hint: Start by computing the asymptotic behaviour of E(e, η) and F(e, η) for η → 0. This can be achieved by expanding in series of η the integrands in Eqs. (13)-(14) (the first non-trivial term will suffice) and then performing the integration. Then, use the resulting asymptotic expressions in Eqs. (10)-(12); there, please remember to appropriately expand in series of η also all the other terms that depend on η.

3 Gravitational potential energy

(a) Recall (from W4 lectures) that the total (scalar) gravitational potential energy of a self-gravitating fluid system is defined by

By substituting in the above equation the expression of the internal gravitational potential given at Eq. (6), show that, for a general ellipsoid, the total gravitational potential energy is given by

where

(b) Compute the explicit expression of the total gravitational energy of a uniform. sphere of radius a by considering spherical limit (i.e., e → 0, η → 0) of Eq. (16).

4 Dynamics

A rigidly rotating ellipsoid is considered to be a figure of equilibrium if its shape does not change over time, i.e. there is the appropriate balance between gravitational, centrifugal, and pressure forces. Now, consider a frame. of reference that co-rotates with the ellipsoid. By requiring that the surface of the ellipsoid introduced at Eq. (1) is an equipotential surface for the effective gravitational potential

which is computed in such co-rotating frame, then the ellipsoid will be a figure of equilibrium. Note that there exist several families of ellipsoids that satisfy such a requirement!

(a) By virtue of the triaxial symmetry of the surface introduced at Eq. (1), it is sufficient to evaluate Eq. (18) at the termination points of the semi-major axes: p1 = (a1, 0, 0), p2 = (0, a2, 0), p3 = (0, 0, a3). Therefore, by evaluating

show that they correspond to the following conditions

(b) Now, consider the oblate case (i.e. η = 0). Show that Eq. (21) reduces to A2 = A1 (as expected from the axisymmetry of the oblate case) and, by considering the limit of Eq (22) for η → 0, compute the resulting explicit expression for  as a function of e.

(c) Plot the expression for  as a function of e, as derived in Q(4.b). Compute the asymptotic behaviour of such a function for e → 0 (the first non-vanishing term will suffice) and, for comparison, plot it against the full expression.

Hint: A single figure with two curves will suffice. Remember that e ∈ [0, 1].

(d) Numerically compute the value of the eccentricity emax at which the function , as derived in Q(4.b), attains its maximum, and compute such maximum value too. Please provide the two values to 5 decimal places.

Hint: To find emax, numerically solve the condition for the first derivative of the function to vanish

Historical note: Congratulations, you have just studied the main properties of the famous Maclaurin spheroids! Colin Maclaurin, who was a Professor in our very own School of Mathematics from 1725-1746, invented (or discovered?) this family of equilibria while he was trying to mathematically determine the shape of the Earth, which was a highly topical problem at that time. These spheroids have later found several other areas of applications in mathematical physics, from rotating water droplets to rotating stars and elliptic galaxies. This is only one example of several other families of ellipsoidal figures of equilibrium - if you want to know more, read the homonymous book by Chandrasekhar!