Using Kepler's
third law which relates the planet's period with its semi-major axis
we could determine all solar system distances once the size of the astronomical
unit is known. If we could measure the distance to Mars we could determine
the size of the solar system. If we use the earth's diameter as a baseline
we could setup two observers on the earth who are diametrically opposite
to each other. Their measurement of the parallax of Mars could then be
used to obtain the distance to Mars, and hence, deduce the size of the
astronomical unit.
Given that the semi-major axes of Mars and Earth are 1.52AU and 1.00AU
respectively ( 1AU = 150x106 km), and that the diameter
of the Earth is 12,742 km :
a) What configuration of Earth/Mars would give the largest parallax?
b) What is this largest parallactic angle in radians and degrees? (
Note: 1 radian = 57.29 degrees, 1 deg = 60 minutes (60'), one minute = 60 seconds(60''))
c) What is the resolving power of the human eye?
d) What technological requirements are needed for determining this parallax?
Edmund Halley's method and the transit of Venus
original reference Astronomical Unit
local reference for Astronomical Unit
Stellar Parallax - Measuring the distance to stars.
Since the distances to stars are so great, the angles involved are
extremely small. The largest baseline that one can use is the diameter
of the earth's orbit. Hence, a measure of the angle that a star makes with
respect to the "far" distant background at opposites sides of the orbit
can be converted into a distance. The convenient unit of distance in astronomy
is the "parsec". One parsec (pc) is defined to be the distance at which
1 astronomical unit (1AU) subtends an angle of 1 arc second(1''). The astronomical
unit is the mean radius of the earth's orbit.
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Given the following:
1AU = 150 x 109 m g = 1 second of arc ( 1/60/60 degrees) ( Convert to radians!) g( 1 sec arc) = 1/(3600*57.29) = 4.85x10-6 radians Determine the length of a parsec in terms of AU and meters. 1pc = 1AU/4.85x10-6 = 2.06x105 AU 1 pc = 150x109 m /4.85x10-6 = 3.09x1016 m = 3.28 ly |
Modern satellite surveys have measured parallax with extreme precision
and have mapped the distances in the galaxy.
See the results from the Hipparcos
satellite and the new GAIA
satellite.
Local site to compare astrometry
Aberration of Starlight
Robert Hooke ( 1669 ) believed he had seen a stellar parallax in the
motion of a star called gamma draconis of 15" . This turned out to be an
effect due to the finite speed of light and the Earth's orbital motion,
rather than a genuine parallactic shift( James Bradley, 1728).
In panel (A) we imagine a telescope tube of length l attached to the
surface of the Earth, which carries the telescope at a velocity v, to the
left. A ray of starlight enters the telescope at the top and takes a finite
amount of time to reach the image plane. During this time of light transit
the telescope itself has moved a distance d along with the earth. The net
effect is that the ray of light appears to have made an angle g
with respect to the telescopes axis. Suppose the event depicted in panel
(A) occurs when the Earth is at location A in its orbit as depicted in
panel (B). A half year later the earth is at point B moving in exactly
the opposite direction from point A. In this case the ray of light from
the star strikes the image plane a distance d to the left of the telescope
axis. The net apparent shift in angle of the star is then 2g
.
(A)
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(B)
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Calculation of aberration of starlight.
1) Determination of the orbital speed
of the Earth?
a) What is the circumference of
the Earth's orbit?
b) What must be the average orbital
speed | v | ?
2) Call the speed of light c. (c = 3x108 m/s).
a) What is the length of time, t, it takes for the starlight to travel
through the tube of length l?
b) How far has the tube moved during this time, that is, d = ?
c) Given that g
= d/l, determine this angle in terms of v and c.
d) Determine the numerical result
for 2g.
Remember to convert from radians to degrees. How does this compare to Hooke's
value?