Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 1

Choose intrinsic Cartesian coordinates in which the oblate spheroid isLet $\ell$ be distance along the line of sight and let $y$ be the sky coordinate in the plane containing the line of sight and the symmetry axis. A rotation through the inclination $i$ givesSubstitution into the ellipsoid equation and minimization over $\ell$, equivalently requiring the quadratic in $\ell$ to have zero discriminant on the projected boundary, givesTherefore the projected axis ratio of an oblate spheroid isIt correctly gives $Q=1$ face-on and $Q=q$ edge-on.

Uniform orientation on the unit sphere means that probability is proportional to the solid-angle elementBecause the minor axis is unoriented, it is enough to take $0\le i\le \pi /2$. After integration over the azimuth and normalization,Equivalently, the random variable $\mu =\cos i$ has the uniform distribution on $[0,1]$.

At fixed intrinsic ratio $q$, part a givesSince $\mu$ is uniform, the change-of-variables formula for a probability density yields the conditional probability density functionThe inverse-square-root singularity at $Q=q$ is integrable, and direct integration gives one.

For an intrinsic probability density function $\mathcal{P}(q)$, the law of total probability averages the conditional density from part c. An object observed with ratio $Q$ can only have $q\le Q$, soThis is an Abel transform of $\mathcal{P}(q)/\sqrt{1-q^2}$. Its normalization follows by reversing the order of integration and using ${\int }_q^1\mathcal{P}(Q\mid q)dQ=1$.

For the uniform distribution $\mathcal{P}(q)=1$, substitute $q=Q\sin \theta$ into part d:where $K$ is the complete elliptic integral of the first kind. Both $Q$ and $K(Q)$ increase on $(0,1)$; indeed $\mathcal{P}(Q)\sim \pi Q/2$ near zero and diverges logarithmically as $Q\uparrow 1$. Thus the apparent distribution is strongly skewed toward higher $Q$: projection makes many intrinsically flattened systems look round.

Withthe factor $\sqrt{1-q^2}$ cancels from the Abel transform:HenceThis probability density function is normalized and again favors rounder projections.

For an ordinary intrinsic density, inversion of the Abel transform givesIf $\mathcal{P}(Q)=1$, the integral is $\pi /2$ for every $q>0$, so its derivative vanishes. The missing probability is an endpoint atom: all systems must be infinitely thin,Indeed, putting $q=0$ directly into part c gives $\mathcal{P}(Q\mid 0)=1$. Thus a uniform apparent-axis-ratio distribution corresponds to a Dirac delta distribution of ideal zero-thickness disks, rather than to a regular intrinsic density.

Yes. Even under random orientation, a population concentrated at one nonzero intrinsic ratio $q_0$ haswhich decreases from an integrable divergence at the lowest allowed value $Q=q_0$. It is therefore skewed toward the lower end of its support. More generally, a sufficiently narrow intrinsic distribution can retain such low-$Q$ peaks after the mixture in part d.

If the random-orientation assumption is relaxed, preferentially edge-on selection makes $\cos i$ concentrate near zero and hence makes $Q$ concentrate near $q$. Dust extinction, surface-brightness selection, or alignment by environment can instead bias the sample in either direction. Triaxial or prolate galaxy shapes also invalidate the projected axis ratio of an oblate spheroid formula and can produce other low-$Q$ distributions.