BINARYSTARS-
CASE STUDY
Solution
[Physics
of Binary Stars] [Binary
Star Case Study][Case
Study Solution]
Gravitation|Astronomy|Topics
The solution to Stackpole's
problem lies in understanding the Doppler Effect.
For a receding source
then the observed wavelength (l')
is related to the true wavelength
(l)by:
l'=
( 1 + vs
)
l
c
|
Doppler Shift
- Source Receding
|
We readily see that because
fl
= c , then taking logs, lnf + lnl
= lnc and differentiating
since c is constant then df/f
+ dl/l
= 0 . From this we see that
|
df/f
= - dl/l=
- vs/c if vs<< c
|
using the result above
where dl
= l - l'.
.
Inserting the values
given of 2 x 106 ms-1 for vs and 434.05x10-9
m
for l
and using
c = 3x108 ms-1 gives a wavelength shift of dl
= 2.98x 10-9 m. This is the doppler
shift of the hydrogen line for the system as a whole.
Analysis of the splitting
of this spectral line into two requires the analysis of our binary
system's motion. Using the result for the Time Period of rotation:
we see that this reduces
to the simpler
|
Separation 'd'
mass of smaller star 'M'
|
for the system in question.From
this we can deduce that the angular speed (w)
of their
motion is given by w
= 2p/T
and since v=rw
, then the linear speed of the smaller
star, Enealor is given by
where r is the distance
from the centre of Enealor to the centre of mass of the system.
A similar calculation to that in 'Physics
of Binary Stars' yields r = 2d/3.
We thus see that the
linear speed of Enealor relative to the centre of mass of the system
is given by
 |
Linear speed of
Enealor about
Centre of Mass
|
A diagram should
make this clearer:
Inserting the necessary
values into this equation gives us a speed
|
v = 5.16x104
ms-1
|
Linear speed of
Enealor
|
From this we use the approximate
doppler shift again:
|
df/f
= - dl/l=
- vs/c if vs<< c
|
Inserting the values given
yields an additional Doppler Shift of
Thus this is a shift of
2.968 - 2.89 = 0.078nm. Doubling this (for motion toward the observer
to be taken into account) gives a splitting of the Hydrogen spectral
line
of 2x0.078 = 0.16nm.
|
?
|
It now remains to
prove that this spectral shift occurs over a cyclaeof 47 days:
Using our time period
formula for the binary system:
and inserting the values
given we find that Enealor moves about G with a period of 8109530
seconds or approximately 94 days. It is thus moving directly away from
the
earth 47 days after it is moving directly towards it (relative to G).
Binary
Star Physics |
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