Environmental Engineering Reference
In-Depth Information
c
cos
α
γ(u)(
1
−
v
y
=
c
sin
α
)
.
(6.37)
u
−
These two equations imply that
sin
α
−
v
x
v
y
=
u/c
cos
α
1
tan
α
=
1
u
2
/c
2
.
(6.38)
−
Stellar aberration is greatest when
α
=
0, in which case this result simplifies to
u
c
1
tan
α
=−
1
u
2
/c
2
,
−
u
c
.
i
.
e
.
sin
α
=−
Now if
u
c
then this is gives rise to a variation in the star's angular position of
≈
2
u/c
over the course of one year, which is in accord with observations.
Example 6.3.1
Consider three galaxies, A, B and C. An observer in A measures
the velocities of B and C and finds they are moving in opposite directions each
withaspeedof
0
.
7
c. (a) At what rate does the distance between B and C increase
according to A? (b) What is the speed of A observed in B? (c) What is the speed of
C observed in B?
Solution 6.3.1
Again it really helps to draw a picture: we refer to Figure 6.11.
(a) The relative speed between B and C according to A is just
2
u
1
.
4
c.Wedo
not of course worry that this speed is in excess of c because it is not the speed of
any material object. (b) According to B, A moves 'to the right' with speed u.(c)
Now to determine the speed of C according to an observer in B we do need to use
the addition of velocities formula since we only know the speed of C in A and the
speed of A relative to B. In classical theory, the result would be
1
.
4
c, but this will
clearly be modified to a value smaller than c in Special Relativity. The correct value
is found using Eq. (6.33):
=
+
u
u
1
.
4
c
1
.
49
=
=
0
.
94
c.
u
2
/c
2
1
+
u
A
u
B
C
Figure 6.11
Relative motion of three galaxies viewed from an observer in A.
