Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S30

A exoplanet transit occurs when a planet crosses the projected stellar disk and blocks some stellar flux. For a circular orbit with isotropically distributed angular-momentum directions, $\cos i$ is uniform on $[0,1]$. A transit requires $a|\cos i|<R_{\star }+R_p$, soFor a dark planet, a uniformly bright stellar disk and negligible dilution, the transit exoplanet transit depth is $\boxed{\delta \simeq (R_p/R_{\star }{)}^2}$. For $R_{\star }/a\ll 1$, a small planet and impact parameter $b=a\cos i/R_{\star }$,The central-transit result sets $b=0$. A first-to-fourth-contact duration retaining planet size replaces $\sqrt{1-b^2}$ by $\sqrt{(1+R_p/R_{\star }{)}^2-b^2}$ and includes the exact projected-speed geometry when needed. Eccentricity, limb darkening and grazing geometry modify these expressions.

The common one-per-cent exoplanet transit depth gives $R_p=0.1R_{\star }$. Apply the supplied main-sequence mass-radius relation, $R_{\star }/R_{\odot }=M_{\star }/M_{\odot }$, and Kepler's third law, neglecting the planet mass. For $P=6$ days,The companions are approximately Jupiter-sized and Earth-sized respectively; these are sizes, not independently determined masses. The duration uses the small-planet chord approximation. Counting the finite planet radius would increase first-to-fourth-contact durations by about ten per cent for central transits.

For equal distance, passband, cadence and noise dominated by stellar photon counting, the supplied mass-luminosity relation makes the solar-mass host about $10^3$ times more luminous, with a transit about $4.64$ times longer. At equal fractional exoplanet transit depth, signal-to-noise per event scales as $\delta \sqrt{{\dot{N}}_{\gamma }{t}_{\mathrm{tr}}}$, favouring the solar-mass host by about $\sqrt{1000\times 4.64}\simeq 68$. Equal apparent brightness would remove the luminosity advantage but retain the duration advantage. Without distances and noise properties the relative detectability is conditional. Transit probability is also larger for the solar-mass case, about $7.2\%$ versus $1.55\%$.

To distinguish the companion, measure the host's radial velocity reflex motion to infer a mass; a low-mass star generally produces a much larger signal than a planet. Also test for a detectable secondary eclipse, colour-dependent dilution, doubled spectral lines and out-of-eclipse ellipsoidal variation. High angular-resolution imaging and shifts of the image centroid can identify a blended background eclipsing binary. A Jupiter-sized radius alone is not decisive because low-mass stars and brown dwarfs can have comparable radii. The three labelled effects on the transit are treated individually below.

Reducing orbital inclination from exactly edge-on increases the impact parameter and shortens the chord, giving $t_{\mathrm{tr}}\propto \sqrt{1-b^2}$ in the small-planet, circular approximation. For a uniformly bright disk, a fully internal transit retains approximately the same exoplanet transit depth $(R_p/R_{\star }{)}^2$. Once grazing, the occulted area and exoplanet transit depth decrease; for $b>1+R_p/R_{\star }$ the exoplanet transit disappears. With limb darkening, a higher-$b$ path is also typically shallower before grazing.

Limb darkening makes the central stellar surface brighter than the disk mean. A small planet at projected position $\mathbf{x}$ blocks fractional flux approximately $\delta (\mathbf{x})=(R_p/R_{\star }{)}^{2}I(\mathbf{x})/⟨I⟩$. A central transit is therefore deepest near mid-transit and rounded rather than flat-bottomed; a near-limb chord can be shallower. The geometrical contact times do not change for fixed radii and orbit, although a threshold-based measured duration can change because ingress and egress are fainter.

As the star expands onto the red giant branch, a planet of fixed radius produces a shallower exoplanet transit, $\delta \propto R_{\star }^{-2}$. For a fixed circular orbit and an approximately central chord, duration grows approximately as $R_{\star }$, and transit probability also rises. At fixed orbital inclination the impact parameter in stellar-radius units decreases, further changing the chord. These statements neglect orbital evolution, stellar mass loss, tidal effects and eventual engulfment, all of which can matter for a very close planet.

The habitable zone is the range of orbital distances where an appropriate planetary atmosphere could sustain liquid water at the surface; it is not a guarantee of inhabited or even habitable conditions for every planet. At a fixed habitable incident flux, $a_{\mathrm{HZ}}\propto L_{\star }^{1/2}$. With the supplied $L_{\star }\propto M_{\star }^3$, Kepler's third law givesTaking Earth as a rough central reference, a $0.1M_{\odot }$ star has $a_{\mathrm{HZ}}\sim 0.0316\mathrm{AU}$ and $P_{\mathrm{HZ}}\sim 6.5$ days. Its six-day, Earth-sized companion receives about $1.11$ times terrestrial irradiance and could therefore lie in the habitable zone in this simplified model. Atmosphere, stellar activity, tidal locking and volatile retention remain unknown. The six-day companion around the Sun-like star receives about $239$ times terrestrial irradiance and is approximately Jupiter-sized; it is not a plausible surface-water Earth analogue. The stellar spectral dependence of actual habitable-zone edges is omitted by the simple scaling.

A cold, self-gravitating region in a molecular cloud can collapse when gravity overcomes thermal, turbulent and magnetic support. Increasing central density produces a protostar, while radiation removes part of the released gravitational energy. The cloud initially carries angular momentum; material that cannot fall radially to the centre settles into a rotating protoplanetary disk. Collisions and cooling reduce random vertical motion, while angular-momentum transport allows some material to accrete and some to move outwards. Bipolar outflows and magnetic stresses can remove additional angular momentum.

The schematic sequence is $\text{rotating cloud}\to \text{protostar plus disk}\to \text{accreting solids and gas}\to \text{planetary system plus cleared disk}$. In the cool disk, grains condense and collide; settling and concentration can produce planetesimals, whose gravitational growth leads to planetary embryos. Massive cores may capture gas before the disk disperses; sufficiently massive unstable disks can also form bound condensations. Migration and interactions modify the final orbits. Radiation, accretion and outflows eventually disperse the remaining gas. Thus planet formation uses the same rotating reservoir produced by ordinary star formation, although the efficiency and final architecture are not universal and individual growth stages remain active research topics.

Neglect disk self-gravity and take $|z|\ll r$ around a central point mass $M_{\star }$. Its vertical gravitational acceleration isVertical hydrostatic equilibrium is $dP/dz=\rho g_z$. For a vertically isothermal ideal gas, $P=\rho c_s^2$ and $c_s$ is independent of $z$, givingHere $v_K=r\Omega =\sqrt{G{M}_{\star }/r}$, $c_s^2=k_{B}T/(\mu m_p)$ is the isothermal sound speed squared, and $h$ is the Gaussian scale height. A different convention writing $e^{-z^2/h^2}$ changes the quoted height by $\sqrt{2}$ but not the physics.

At $T=100\mathrm{K}$ with $\mu =2.3$, $c_s=\sqrt{k_{B}T/(\mu m_p)}\simeq 599\mathrm{m}{\mathrm{s}}^{-1}$. A solar-mass Keplerian disk has $v_K\simeq 29.8\mathrm{km}{\mathrm{s}}^{-1}$ at $1\mathrm{AU}$, so $\boxed{h/r\simeq 0.020}$: the thin-disk approximation is consistent.

For fixed central mass, $h/r\propto T^{1/2}{r}^{1/2}$. If $T\propto r^{-q}$, thenA flared outer disk intercepts more stellar radiation than a flat or self-shadowed disk, affecting its temperature, chemistry, vertical structure and observed spectral energy distribution. Heating changes the sound speed and therefore transport and dispersal; it can also enhance irradiation-driven mass loss. The actual coupled evolution depends on optical properties and transport, so flaring alone does not fix a unique lifetime. “Mid-planet” in the source is read as mid-plane.

The planetary equilibrium temperature is the effective planetary temperature at which absorbed starlight balances emitted thermal radiation, neglecting internal power. With uniform reradiation over the full surface,Cancelling areas givesUsing the supplied Bond albedos and taking the thermal emissivity $ϵ=1$ yields $\boxed{T_{\mathrm{eq},\oplus }\simeq 255\mathrm{K},T_{\mathrm{eq},\mathrm{J}}\simeq 103\mathrm{K}}$. “Perfect black bodies” must here mean perfect infrared emitters, since a perfect absorber at all wavelengths would instead have $\alpha =0$, inconsistent with retaining the supplied reflected fractions. If both albedos were set to zero, the corresponding values would be about $279$ and $122\mathrm{K}$.

Earth's mean surface temperature of about $288\mathrm{K}$ exceeds its emission temperature mainly because of the greenhouse effect. Jupiter also radiates substantial internal energy from cooling and contraction, including interior differentiation processes, so absorbed sunlight alone underpredicts its emission. Jupiter has no solid surface; the quoted $150\mathrm{K}$ is a representative atmospheric level and is not identical to an effective radiating temperature.

At full phase, reflected-light contrast iswhere $A_g$ is the geometric albedo, and other phases multiply this by a phase function normalized to one at full phase. Thermal contrast for uniform blackbody emitters isThe corresponding bolometric thermal ratio is $(R_p/R_{\star }{)}^2(T_p/T_{\mathrm{eff}}{)}^4$. If zero internal heating and equilibrium are assumed, energy balance also gives a bolometric reflected ratio $\alpha (R_p/d{)}^2/4=(\alpha ϵ/(1-\alpha ))(R_p/R_{\star }{)}^2(T_p/T_{\mathrm{eff}}{)}^4$ for the luminosity ratio; it is not a directional full-phase contrast.

Earth's thermal emission peaks around $10\mu \mathrm{m}$, and solar emission around $0.5\mu \mathrm{m}$. At $0.5\mu \mathrm{m}$ the thermal contrast for $T_p=288\mathrm{K}$ is negligible, about $5\times 10^{-46}$; reflected light dominates. Taking illustrative visible $A_g\simeq 0.3$ gives $\boxed{C_{0.5}\sim 5\times 10^{-10}}$. A Bond albedo does not uniquely determine a spectral geometric albedo; a Lambert sphere with Bond albedo $0.3$ would instead have $A_g=0.2$, giving $3.6\times 10^{-10}$. At $20\mu \mathrm{m}$, the Planck function gives $\boxed{C_{20}\simeq 1.0\times 10^{-6}}$ for $288\mathrm{K}$; using $255\mathrm{K}$ gives about $7\times 10^{-7}$. These are order estimates, since the terrestrial spectrum is not a uniform blackbody.

At maximum projected separation, an Earth analogue at $10\mathrm{pc}$ is separated from its star by $\vartheta =a/D_{\mathrm{sys}}=0.1\mathrm{arcsec}=4.85\times 10^{-7}\mathrm{rad}$. The Rayleigh criterion for a circular pupil givesThe relevant quantity is the star–planet separation, not the tiny angular diameter of the planet itself. Simple angular optical resolution is insufficient: the planet lies in the star's diffraction pattern at reflected contrast of order $10^{-10}$, or thermal contrast of order $10^{-6}$. Residual speckles, wavefront errors, scattered light, detector/background noise and, in the infrared, strong telescope and sky thermal emission obstruct detection. A coronagraph or other stellar suppression method generally has an inner working angle of several $\lambda /D$, making practical aperture requirements larger than the bare Rayleigh criterion. The question's instrumentation claim is interpreted in its 2019 context; the physical requirements remain the basis of the estimate. Maximum reflected brightness near full phase also coincides with small projected separation, so maximum brightness and maximum resolvable separation cannot generally be obtained simultaneously.