19 Aberration
217
:
9
•
•
•
•
?
?
A(t 1 )
B(t 1 )
C(t 2 )
T
O
-
6
6
-
×
x
x
v
S
α
a = a
γ
a = a
γ
b = b
b = b
t 2 t 1
Σ
Σ o
Fig. 19.2 Observation of the star in
AC O
u l = c L T l = 2b +
(a − β(u l − 2b))
2
+ b 2 .
(244)
With (242), we get for (243)
u r =
(Lγcosα + βu r ) 2 + L 2 sin
2
α .
(245)
This is a quadratic equation for u r with the solution
u r =
L
γ
(β cos α
+ 1) .
(246)
The second solution (L/γ) (β cos α
− 1) has no physical meaning, since it would
be a negative distance.
With (242), we get for (244)
u l − 2L sin α
=
Lγ cos α − β(u l − 2L sin α )
2 + L 2 sin
2
α ,
(247)
which yields the quadratic equation
u
2
l γ
2
− u l 2L(2 sin α
γ
2
− βγ cos α
)
+ 4L
2
γ
2 sin
2
α
− L
2
(γ
2 cos
2
α
+ sin
2
α
) − 4L
2
βγ cos α
sin α
= 0
with the formal solutions