Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S2

For the two-body problem, the centre of mass lies a distance $a{M}_J/(M_{\odot }+M_J)=a/1048$ from the Sun. With $a=5.2\mathrm{AU}$,The relative circular-orbit period is $P=2\pi \sqrt{a^3/[G(M_{\odot }+M_J)]}\simeq 11.85$ years. The Sun's orbital speed about that centre of mass is thereforeThis is the space-speed amplitude; the observed radial velocity amplitude also includes the orbital inclination.

Kepler's third law is $P^2=4{\pi }^2{a}^3/[G(M_S+M_P)]$, with $a$ the semimajor axis of the relative orbit. For a circular edge-on orbit, the star's speed is $V_S=2\pi a_S/P$, and the centre of mass relation gives $a_S=a{M}_P/(M_S+M_P)$. Cubing the speed expression and eliminating $a$ yieldsFor $M_P\ll M_S$, this reduces toWithout the edge-on assumption, a factor ${\sin }^{3}i$ multiplies the numerator, explaining the minimum-mass degeneracy of the radial-velocity method.

Converting luminosity to mass needs a main-sequence mass-luminosity relation. Using the near-solar approximation $L/L_{\odot }\simeq (M/M_{\odot }{)}^4$ gives $M_S\simeq 0.946M_{\odot }$. ThenFor $P=11.6$ days and $V_S=10\mathrm{m}{\mathrm{s}}^{-1}$, these give approximatelyThese estimates retain the circular, edge-on and low-companion-mass assumptions. Adopting a mass-luminosity exponent $3.5$ instead changes the quoted values by about one percent; luminosity alone does not determine a unique stellar mass without such a calibration.

The effective temperature is defined by the Stefan–Boltzmann law applied to a star's total luminosity:Here $R$ is the adopted photospheric radius and ${\sigma }_{\mathrm{SB}}$ is the Stefan-Boltzmann constant. This defines the temperature of a black body radiating the same total flux per unit surface area; it does not assert that every wavelength of the stellar spectrum follows a perfect Planck spectrum.

The parallax gives $d=1/0.00763\simeq 131\mathrm{pc}$. Ignoring extinction, the distance modulus givesThe bolometric magnitude relation then yields $L/L_{\odot }=10^{0.4(4.72-M_{\mathrm{bol}})}\simeq 2.53\times 10^4$. With $T_{\odot }\simeq 5772\mathrm{K}$,A star this cool and luminous belongs to the red-supergiant region of the Hertzsprung-Russell diagram. Extinction and the adopted effective temperature are important systematic inputs; this calculation uses the supplied temperature to infer the radius.

A first distance-independent method fits the continuum shape or a calibrated colour index to an atmosphere model. A black-body approximation uses Wien's displacement law, ${\lambda }_{\max }T\simeq 2.90\times 10^{-3}\mathrm{m}\mathrm{K}$, or ratios of the Planck spectrum at two wavelengths. It requires a known extinction correction and sufficient treatment of molecular absorption and line blanketing; a cool supergiant need not have a black-body colour temperature equal to its effective temperature.

A second method fits temperature-sensitive spectral lines or molecular bands. Boltzmann excitation and the Saha equation explain why excitation and ionization ratios change with temperature; detailed atmosphere calculations also include surface gravity, composition and departures from local thermodynamic equilibrium. TiO molecular bands are particularly useful for cool stars. Line ratios remove the overall distance normalization, but their calibration remains model dependent.

The direct flux-radius method can be made distance independent as well: an independently measured angular diameter $\Theta$ gives $T_{\mathrm{eff}}=[4F_{\mathrm{bol}}/({\sigma }_{\mathrm{SB}}{\Theta }^2){]}^{1/4}$. For a large nearby supergiant with a resolved photosphere, good bolometric flux and a limb-darkened interferometric angular diameter, this is often the most direct determination. Its advantage depends on those measurement uncertainties and a consistent radius definition. The radius inferred in the preceding part cannot itself provide an independent temperature measurement, because the supplied temperature was already used to calculate it.

The virial theorem for a stationary isolated system with inverse-square gravity iswhere $K$ is total kinetic energy in the system's centre-of-mass frame and $W$ is its negative gravitational potential energy. The brackets denote a suitable time or ensemble average. Surface pressure, rapid evolution or mass fluxes add terms, so equilibrium and an appropriate boundary are substantive assumptions.

For an isotropic velocity dispersion, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$. To specify the order-unity structure factor, approximate the gravitating system by a uniform sphere, whose gravitational potential energy is $W=-3G{M}^2/(5R)$. The virial theorem givesThe numerical value uses $R=1\mathrm{Mpc}$ and ${\sigma }_{\mathrm{los}}=900\mathrm{km}{\mathrm{s}}^{-1}$. Different mass profiles change the prefactor, rather than the scaling $R{\sigma }^2/G$.

For fully ionized intracluster gas, take mean particle mass $\mu m_p$ with $\mu \simeq 0.6$. In the same uniform-sphere virial estimate, replace ${\sigma }^2$ by $k_{B}T/(\mu m_p)$ to obtainAlternatively, hydrostatic equilibrium gives $M(<R)=-k_{B}TR(G\mu m_p{)}^{-1}(d\log {\rho }_g/d\log r+d\log T/d\log r)$. An isothermal gas with density proportional to $r^{-2}$ gives $1.3\times 10^{15}{M}_{\odot }$. The supplied radius and temperature do not specify the profile, so the meaningful inference is a mass of order $10^{15}{M}_{\odot }$, with the displayed model choices making the coefficient explicit. The high supplied temperature need not yield the same estimate as the galaxy dispersion in a crude single-profile calculation.

The stellar estimate is much smaller:It counts luminous stellar matter, rather than all matter in the cluster.

Nonbaryonic dark matter can provide the additional gravitating mass. Gravitational lensing independently maps cluster mass, while the cosmic microwave background and Big Bang nucleosynthesis constrain the cosmic matter and baryon budgets. The separation of lensing mass from strongly interacting X-ray gas in merging clusters is evidence that much of the gravitating component is not that hot gas.

Uncounted baryonic matter, including the observed hot gas, faint stars and compact remnants, certainly adds to the stellar estimate. X-ray emission directly establishes a large gas reservoir. However, baryon-budget constraints and microlensing constraints on compact objects prevent arbitrarily filling the entire discrepancy with ordinary hidden objects.

Modified gravity could change the inference from a measured acceleration to a Newtonian mass. Such a theory must also reproduce the observed lensing and cosmological evidence; explaining some rotation curves alone is insufficient, and cluster mass discrepancies remain a demanding test. Departures from equilibrium and projection can bias individual mass estimates, but would have to explain the independent observations systematically rather than merely modify one virial prefactor.