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Marian Physics!
Parallax
Flat Prairie Publishing
Parallax
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Measuring the distance to a
star is a pretty difficult
task. Direct measurement
using a meter stick or long
tape measure is not
possible. For stars that are
close to us parallax used.
Section 1
Parallax
Parallax:
One of the most difficult problems in astronomy is determining
the distances to objects in the sky. There are five basic methods of determining distances: radar, parallax, cepheid variables,
standard candles, and the Hubble Law.
Each of these methods is most useful at certain distances, with
radar being useful nearby (e.g., the Moon), the Hubble Law being useful at the farthest distance (e.g., galaxies far, far away).
In this exercise, we investigate the use of the measured parallax method to determine distances to nearby stars, those within
about 650 light years from the Sun.
Even when observed with the largest telescopes, most stars
are still just points of light. Although we may be able to tell a lot
about a star through its light, these observations do not give us
a reference scale to use to measure its distance. We need to
rely on a method that you are familiar with: the parallax of an object.
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Extend your arm directly away from your body. Now take one
finger and point it at a distant object. Alternately close each
eye. When you close one eye the position of your finger with respect to the distant object will change. This is parallax. This is
because the centers of your eyes are 5 – 6 centimeters apart,
so each eye has a different point of view. As a side note most
people have a dominant eye. If you close your left eye and your
finger does not move, your right eye is dominant and verse
visa. Since the earth is in orbit around the sun, we can look at a
nearby star six months apart and measure the angle. This angle is called the parallax angle.
The further the star is away from us the smaller the parallax angle.
the angle is large
The actual parallax angle is the angle as indicated below.
This allows us to use simple trig to calculate the distance to a
star. As you can see the opposite side to the parallax angle is
1AU (Astronomical Unit, the average distance from the earth to
the sun). The sin function will then give us the distance to the
star.
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Of course astronomers are not ones to use simple methods.
They have goobered up this measurement just like the ones
with magnitude. So keep reading and the enumeration as to
how astronomers use parallax to measure the distance to a star
will be given.
Except for our sun, the stars are pretty far away. Because of
this the parallax angle is extremely small. Astronomers use arc
seconds (one second of one minute of one degree of angle).
The arc second is 1/3600 of a degree or .00103 degrees. Take
a protractor and try to draw 3600 lines between each degree
mark. Now you know how hard that small of an angle is to
measure.
Instruments that could measure such a small angle were not
available until the mid 1800’s. Around 1830, Thomas Henderson, working in South Africa, Friedrick Wilhelm Struve, and Friedrick Wilhelm Bessell (both in Europe) started working in earnest on measuring the distance to a star.
Henderson actually got the jump on the others, making measurements of Alpha Centauri. In 1833, he packed up and went
back to England, along with his data. He was in no hurry to reduce his data, so it languished for years. When he finally did
get to looking at his measurements, he found that there did
seem to be what may have been a parallax shift in Alpha Centauri, but he did not trust his data. He had only 19 measurements and the instrument that he had been using had been
damaged in shipping to South Africa. He decided to wait for better measurements.
As it turns out, Henderson’s data were approximately correct.
However, he didn’t know that, so he held off publishing his findings until he had more data.
In 1837, Struve announced that he had measured the distance
to Vega. He reported a parallax angle of 0.125 arc seconds.
Bessel was working on measuring the parallax angle for 61
Cygni. He measured a parallax angle of 0.314 arc seconds and
reported it in 1838.
Now we should say the Struve was the first to measure the parallax angle of a star, but most authorities give that honor to Bessel. Why, well read on. In 1938, Struve was convinced that he
had made an error in his first measurement to Vega and reported a parallax angle twice that of his first report. Everyone
was more than a little concerned over the validity of Struve’s results and Struve himself indicated that Bessel was first with a
believable parallax angle. It turns out that Struve was close to
the actual parallax angle to Vega with his first paper and that
the error was in his second paper. So it goes. As a side not
Struve had 12 children by his first wife. After she died he married Johanna Bartels, the daughter of a prominent mathematician. He had six more children with Johanna.
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Parallax the Astronomers Way:
3.26 Light Years (ly)
Astronomers measure the parallax angle in arc seconds. One
of the amazing things about trig is that for very small angles the
tangent of an angle in radians is equal to the measure of the angle in radians. So through some algebraic legerdemain the astronomers come up with a simple equation as follows. (the
sweat and bother section after this will explain where all of this
comes from).
3.086x1013 km
An Example:
The star alpha Centauri has a parallax angle of 0.742 arc seconds. What is the distance to alpha Centauri in parsecs and in
light years.
Where d is the distance to the star in a unit called “Parsecs”
and p is the parallax angle in arc seconds.
Parsecs:
A new fundamental unit of distance in Astronomy
"A star with a parallax of 1 arcsecond has a distance of 1 Parsec."
1 parsec (pc) is equivalent to:
206,265 AU
Some problems:
1. Using Struve’s reported parallax angle of 0.125 arc seconds
to Vega, how far is Vega in parsecs and in light years?
2. Using Bessels reported value of 0.314 to 61 Cygni how far is
it to 61 Cygni in parsecs?
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3. The smallest parallax angle that can be accurately measured
from the ground is 0.01 arc seconds. If the angle is smaller
than 0.01 arcseconds we do not have a protractor good
enough to measure the angle. You measure a star to have a
parallax angle of 0.01 arc seconds. How far away is that star
in par secs and in light years? (Any star that more than this
distance away from us we cannot use ground based parallax
measurements to measure the distance to that star).
4. The Hipparcos satellite measures precision parallax angles
to about 0.001 arc seconds. Hipparcos measures the parallax angle to a star as 0.001 arc seconds, how far away is that
star? Hipparcos has given us parallax measurements on
more than 100,000 stars. (this is the current limit to our measurements using parallax).
5. NASA had proposed a Space Interferometry Mission (SIM)
which would measure parallax angles to .000004 arc seconds. What would be the furthest star that could be measured by this instrument? After spending almost $200 million
on the instrumentation for the mission, the project was cancelled in 2010.
of a human hair at 600 miles. What will be the furthest star
that can be measured with this instrument?
Sweat and Bother:
Those of you who are thinking are now wondering how we can
measure an angle, take its inverse and wind up with a distance
and what the heck are parsecs anyway.
So we start with the right triangle
The tangent of angle p is opposite over adjacent so:
6. In 2013 the European Space Agency plans on launching
Gaia. This satellite will be able to measure parallax angles to
0.00001 arc seconds. The plans are for it to measure the parallax angle to 1,000 million stars. It can see stars down to
magnitude 15. That is the same as measuring the diameter
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Now, the small angle approximation tells us that the tangent of
an angle is equal to the angle for very small angles and the angle is measured in radians. So:
The tangent of p is equal to p for small angles if we measure p
in radians, but we want to measure in arc seconds. We have to
then convert radians to arc seconds. If I measure p arcseconds,
how many radians is that.
(7)
Now, astronomers, like most scientists are lazy individuals they
then simplify.
So they define the parsec such that it makes all the numbers go
away in the equation above. This leaves the simple equation.
Ok some algebra:
with p as the parallax angle measured in arcseconds
but p(radians) is calculated from the measured p in arc seconds
by equation 7 above.
and d will be the distance in parsecs. You have to remember
that when these things were done in the 1800’s we did not have
calculators so doing the proper calculations using tangents etc
would be difficult. Using this method makes the calculations easier but provides one messy derivation.
Combining everything we get.
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