Superluminal motion

Derivation of the apparent velocity

A relativistic jet coming out of the center of an active galactic nucleus is moving along AB with a velocity v, and is observed from the point O. At time ${\displaystyle t_1}$ a light ray leaves the jet from point A and another ray leaves at time ${\displaystyle t_2=t_1+\delta t}$ from point B. An observer at O receives the rays at time ${\displaystyle t_1^{\mathrm{\prime }}}$ and ${\displaystyle t_2^{\mathrm{\prime }}}$ respectively. The angle ${\displaystyle \varphi }$ is small enough that the two distances marked ${\displaystyle D_L}$ can be considered equal.

${\displaystyle AB=v\delta t}$

${\displaystyle AC=v\delta t\cos \theta }$

${\displaystyle BC=v\delta t\sin \theta }$

${\displaystyle t_2-t_1=\delta t}$

${\displaystyle t_1^{\prime }=t_1+\frac{D_L+v\delta t\cos \theta }{c}}$

${\displaystyle t_2^{\prime }=t_2+\frac{D_L}{c}}$

${\displaystyle \delta t^{\prime }=t_2^{\prime }-t_1^{\prime }=t_2-t_1-\frac{v\delta t\cos \theta }{c}=\delta t-\frac{v\delta t\cos \theta }{c}=\delta t(1-\beta \cos \theta )}$, where ${\displaystyle \beta =v/c}$

${\displaystyle \delta t=\frac{\delta t^{\prime }}{1-\beta \cos \theta }}$

${\displaystyle BC=D_L\sin \varphi \approx \varphi D_L=v\delta t\sin \theta \Rightarrow \varphi D_L=v\sin \theta \frac{\delta t^{\prime }}{1-\beta \cos \theta }}$

Apparent transverse velocity along ${\displaystyle CB}$, ${\displaystyle v_{\text{T}}=\frac{\varphi D_L}{\delta t^{\prime }}=\frac{v\sin \theta }{1-\beta \cos \theta }}$

${\displaystyle {\beta }_{\text{T}}=\frac{v_{\text{T}}}{c}=\frac{\beta \sin \theta }{1-\beta \cos \theta }.}$

The apparent transverse velocity is maximal for angle (${\displaystyle 0<\beta <1}$ is used)

${\displaystyle \frac{\mathrm{\partial }{\beta }_{\text{T}}}{\mathrm{\partial }\theta }=\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }\left[\frac{\beta \sin \theta }{1-\beta \cos \theta }\right]=\frac{\beta \cos \theta }{1-\beta \cos \theta }-\frac{(\beta \sin \theta {)}^2}{(1-\beta \cos \theta {)}^2}=0}$

${\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta {)}^2=(1-\beta \cos \theta )(\beta \sin \theta {)}^2}$

${\displaystyle \Rightarrow \beta \cos \theta (1-\beta \cos \theta )=(\beta \sin \theta {)}^2\Rightarrow \beta \cos \theta -{\beta }^2{\cos }^2\theta ={\beta }^2{\sin }^2\theta \Rightarrow \cos {\theta }_{\text{max}}=\beta }$

${\displaystyle \Rightarrow \sin {\theta }_{\text{max}}=\sqrt{1-{\cos }^2{\theta }_{\text{max}}}=\sqrt{1-{\beta }^2}=\frac{1}{\gamma }}$, where ${\displaystyle \gamma =\frac{1}{\sqrt{1-{\beta }^2}}}$

${\displaystyle \therefore {\beta }_{\text{T}}^{\text{max}}=\frac{\beta \sin {\theta }_{\text{max}}}{1-\beta \cos {\theta }_{\text{max}}}=\frac{\beta /\gamma }{1-{\beta }^2}=\beta \gamma }$

If ${\displaystyle \gamma \gg 1}$ (i.e. when velocity of jet is close to the velocity of light) then ${\displaystyle {\beta }_{\text{T}}^{\text{max}}>1}$ despite the fact that ${\displaystyle \beta <1}$. And of course ${\displaystyle {\beta }_{\text{T}}>1}$ means that the apparent transverse velocity along ${\displaystyle CB}$, the only velocity on the sky that can be measured, is larger than the velocity of light in vacuum, i.e. the motion is apparently superluminal.

History

The apparent superluminal motion in the faint nebula surrounding Nova Persei was first observed in 1901 by Charles Dillon Perrine. "Mr. Perrine's photograph of November 7th and 8th, 1901, secured with the Crossley Reflector, led to the remarkable discovery that the masses of nebulosity were apparently in motion, with a speed perhaps several hundred times as great as hitherto observed." "Using the 36-in. telescope (Crossley), he discovered the apparent superluminal motion of the expanding light bubble around Nova Persei (1901). Thought to be a nebula, the visual appearance was actually caused by light from the nova event reflected from the surrounding interstellar medium as the light moved outward from the star. Perrine studied this phenomenon using photographic, spectroscopic, and polarization techniques."

Superluminal motion was first observed in 1902 by Jacobus Kapteyn in the ejecta of the nova GK Persei, which had exploded in 1901. His discovery was published in the German journal Astronomische Nachrichten, and received little attention from English-speaking astronomers until many decades later.

Following a 1939 paper by Couderc, in 1966, Martin Rees pointed out that "an object moving relativistically in suitable directions may appear to a distant observer to have a transverse velocity much greater than the velocity of light". In 1969 and 1970 such sources were found as very distant astronomical radio sources, such as radio galaxies and quasars, and were called superluminal sources. The discovery was the result of a new technique called Very Long Baseline Interferometry, which allowed astronomers to set limits to the angular size of components and to determine positions to better than milli-arcseconds, and in particular to determine the change in positions on the sky, called proper motions, in a timespan of typically years. The apparent speed, which could be up to six times the speed of light, is obtained by multiplying the observed proper motion by the distance.

In the introduction to a workshop on superluminal radio sources, Pearson and Zensus reported

The first indications of changes in the structure of some sources were obtained by an American-Australian team in a series of transpacific VLBI observations between 1968 and 1970 (Gubbay et al. 1969). Following the early experiments, they had realised the potential of the NASA tracking antennas for VLBI measurements and set up an interferometer operating between California and Australia. The change in the source visibility that they measured for 3C 279, combined with changes in total flux density, indicated that a component first seen in 1969 had reached a diameter of about 1 milliarcsecond, implying expansion at an apparent velocity of at least twice the speed of light. Aware of Rees's model, (Moffet et al. 1972) concluded that their measurement presented evidence for relativistic expansion of this component. This interpretation, although by no means unique, was later confirmed, and in hindsight it seems fair to say that their experiment was the first interferometric measurement of superluminal expansion.

In 1994, a galactic speed record was obtained with the discovery of a superluminal source in the Milky Way, the cosmic x-ray source GRS 1915+105. The expansion occurred on a much shorter timescale. Several separate blobs were seen to expand in pairs within weeks by typically 0.5 arcsec. Because of the analogy with quasars, this source was called a microquasar.