Past exam of the astrophysics course of the University of Oxford / 2021 / 3

Work in the source rest frame and neglect the source's small cosmological redshift. Over an emission interval $\Delta t$, a ballistic relativistic jet feature travels $\beta c\Delta t$, advancing $\beta c\Delta t\cos \theta$ toward the observer and $\beta c\Delta t\sin \theta$ across the sky. The second photon has a shorter journey, so its arrival is delayed by onlyDividing the projected displacement by this arrival interval givesApparent superluminal motion is a combination of projection and light-travel delay. The feature's local speed remains $\beta c<c$; its inferred transverse displacement divided by the observer's arrival interval is not a local signal velocity.

For fixed $\beta$, differentiate the apparent speed with respect to angle:The maximum lies at $\cos \theta =\beta$ and has value ${\beta }_{\mathrm{app},\max }=\beta /\sqrt{1-{\beta }^2}=\gamma \beta$. Hence strict apparent superluminal motion requiresThe threshold is an infimum: equality yields exactly ${\beta }_{\mathrm{app}}=1$ at $\theta =\pi /4=45^{\circ }$, rather than a strictly superluminal value. For a chosen larger $\beta$, the optimal inclination is $\theta =\arccos \beta$, and the full superluminal interval satisfies $\sin \theta +\cos \theta >1/\beta$.

For intrinsically symmetric oppositely directed relativistic jets, their proper motions areThe ratio removes $D$ and yieldsTaking the product instead givesCausality requires $\beta \le 1$, so the limiting distance and inclination areHere and in e, “maximum” denotes the limiting $\beta \to 1$ bound; massive ejecta obey strict inequalities. The conversion is $1\mathrm{mas}\mathrm{da}{\mathrm{y}}^{-1}=5.6113\times 10^{-14}\mathrm{rad}{\mathrm{s}}^{-1}$, henceBoth bounds occur together in the light-speed limit. At any specified smaller distance, ${\beta }^2=x^2+(1-x^2)(D/D_{\max }{)}^2$ determines the intrinsic speed.

The Doppler factors for the two discrete blobs are ${\mathcal{D}}_{\pm }=[\gamma (1\mp \beta \cos \theta ){]}^{-1}$. Their ratio is $(1+x)/(1-x)=77/33=7/3$. Lorentz-invariant specific intensity means $I_{\nu }/{\nu }^3$ is unchanged; with $\nu =\mathcal{D}{\nu }^{\prime }$ and the same rest-frame power-law spectrum,The brightness ratio at the common observed frequency therefore givesThe stated identical intrinsic specific intensities must refer to comparable intrinsic states. If the blobs fade with their proper ages, equal observer times sample different ages and need that correction. A steady continuous jet has a different boosting exponent, $2+\alpha$; the discrete ejecta and the specified specific intensity transformation justify $3+\alpha$ here.

For optically thin synchrotron radiation from an isotropic electron distribution $dN/dE\propto E^{-p}$, the characteristic emitted frequency scales as $B{E}^2$ and the total single-particle power as $B^2{E}^2$. Thus $j_{\nu }\propto E^{-p}{E}^2/(d\nu /dE)\propto {\nu }^{-(p-1)/2}$; integration over the full single-particle spectrum preserves this exponent. HenceThe test-particle strong nonrelativistic diffusive shock acceleration prediction is $p=(r+2)/(r-1)=2$ for compression ratio $r=4$, giving $\alpha =0.5$. The inferred spectrum is steeper. If interpreted literally within that uncooled formula, $p=2.50$ would correspond to $r\simeq 3$, but cooling, escape, and other acceleration conditions can also steepen it.

The observed Galactic cosmic ray differential energy spectrum below the knee is approximately $E^{-2.7}$, steeper still; its formal synchrotron radiation equivalent would be $\alpha \simeq 0.85$. Thus $p\simeq 2.50$ lies between the simplest source prediction and the propagated cosmic ray slope. The comparison is qualitative: most measured cosmic rays are nuclei, whereas the radio slope here probes electrons, and propagation and radiative losses alter the spectra differently.

Apply the same symmetric-ejection and causality argument to the launch proper motions, when the proposed ejecta are fastest. One obtainsandThe fitted angular decelerations describe subsequent evolution; they do not relax the bound already imposed at launch. They are observer-time phenomenological coefficients, not directly identical physical decelerations in the source frame.

The stellar parallax distance $2.96\pm 0.33\mathrm{kpc}$ is compatible with the constant-velocity bound $3.435\mathrm{kpc}$, but its central value exceeds the decelerating-launch bound by $0.717\mathrm{kpc}$, or about $2.17$ quoted standard deviations if the uncertainty is treated as symmetric Gaussian. Even its one-standard-deviation lower endpoint, $2.63\mathrm{kpc}$, exceeds $2.243\mathrm{kpc}$. At $D=2.96\mathrm{kpc}$ the respective inferred intrinsic speeds are $\beta =0.885$ and $\beta =1.300$. The latter is a concrete violation of causality under the symmetric model assumptions.

Thus the supplied distance favours the constant-velocity fit. This is a consistency comparison, not a full statistical model selection: uncertainties in the fitted proper motions, their correlations, intrinsic asymmetry and the distance likelihood would be required for that.