 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2021/3/c/solution

For intrinsically symmetric oppositely directed relativistic jets, their proper motions are

μ_{app} = \frac{β c s i n θ}{D ( 1 - β c o s θ )}, μ_{rec} = \frac{β c s i n θ}{D ( 1 + β c o s θ )} .

The ratio removes

D

and yields

x \equiv β cos θ = \frac{μ _{app} - μ _{rec}}{μ _{app} + μ _{rec}} = 0.4.

Taking the product instead gives

μ_{app} μ_{rec} = \frac{c ^{2}}{D ^{2}} \frac{β ^{2} - x ^{2}}{1 - x ^{2}} .

Causality requires

β \leq 1

, so the limiting distance and inclination are

D_{m a x} = \frac{c}{μ _{app} μ _{rec}}, θ_{m a x} = arccos x .

Here and in e, “maximum” denotes the limiting

β \to 1

bound; massive ejecta obey strict inequalities. The conversion is

1 mas da y^{- 1} = 5.6113 \times 1 0^{- 14} rad s^{- 1}

, hence

D_{m a x} = 3.435 kpc, θ_{m a x} = 66.4 2^{\circ} .

Both bounds occur together in the light-speed limit. At any specified smaller distance,

β^{2} = x^{2} + (1 - x^{2}) (D / D_{m a x})^{2}

determines the intrinsic speed.

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