apparent superluminal motion in the
quasar 3C273
1) Attached is
radio map of
the quasar 3C273 showing the major central brightness plus the location
of a
jet as a function of time. Notice
that time is
flowing down the page, and that the locations of the center and the jet
are
given at 5 different times.
The tick marks on the horizontal and vertical scales are separated by
0.002 arc
seconds.
a) Find the apparent
motion of
the jet with respect to the quasar center by recording and measuring
the
angular position of the jet (first in cm, then in arc seconds) at each
time.
b)
Then use the small angle approximation (or trigonometry) to find the
distance
of the jet from the quasar center at each time. The distance to the
quasar is
2.6 x 109 c-yrs.
c)
Find the average speed of the jet as a function of time.
2)
The speed of the jet cannot be superluminal, of course.
a)
Show that the time between 2 such pictures of the quasar/jet as seen by
earth
is given by
DtE = DtQ (1 - v/c cos q)
where v is the
speed of the jet relative to the quasar;
q is the
angle
between the line of sight to earth from the quasar and the velocity of
the jet
relative to the quasar;
and DtQ
is the time between images as seen by the quasar
Hint: draw a picture showing the jet location at two different times
(the
emission of a photon at some time and then the emission of a second
photon at a
later time DtQ ).
Also
show the location of the first photon at the later time.
b) show
that the transverse speed of the jet as seen by earth
is given by
vTE
= v sin q/(1 -
v/c cos q)
c) find
the value of q (in terms of
v/c) that gives the maximum value of vTE
d) show
that this maximum value for vTE is
given by
vTE = v
(1 - v2/c2)-1/2
e) what
is the maximum value of vTE if
v/c = 0.99? v/c = 0.999?
f) Can your
measurements in (1)
be used to determine v/c for the 3C273 jet?
