In 1838 the German astronomer Bessel (Struve and Henderson as well) found that 61 Cygni (a binary system of mag 5.22 and 6.02) shifted, back and forth during the year by 0.29 seconds of arc! This is a measurement that amounts to the thickness of a human hair viewed from the rear of the classroom or a penny seen from a distance of 10.7 km!! Yet this measurement (I have NO idea how he did it but hope to find out) was dramatic. He was awarded the Royal Astronomical Society's Gold Medal in 1841 and Sir John Herschel (Sir William's son) said the following:
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I congratulate you and myself that we have lived to see the great and hitherto impossible barrier to our excursions into the sidereal universe - that barrier against which we chafed so long and so vainly - almost overleaped at three different points. It is the greatest and most glorious triumph which practical astronomy has ever witnessed
Sir John Herschel, 1841
The method of parallax is wonderfully simple. It is illustrated in the following figure (also, click here to see a simulation of stellar parallax):
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"p" is the parallactic angle which is one-half the total amount of shift that one would see in the course of 6 months. The reason that we define p in this way is to make the math very simple. It uses no more complicated math than:
d = rp
where:
The Hipparcos satellite has increased this number dramatically over past decade to about 120 000 stars. Of this number, about 20 000 stars have parallaxes known to an accuracy better than 10%.
Example: The star 61 Cygni
has a parallactic shift of 0.29".
How far away is it?
Solution: Just use r = 1/p. But there's a catch (there's always
a catch!) ... we must use radian measure... so...
| Unit | Conversion |
| 1 arc second | 1/3600 degree |
| n arc seconds | n/3600 degrees |
| 1 degree | p/180 radians |
| n" | (n/3600)(pi/180) radians |
0.29" =( 0.29/3600)(pi/180) = 0.0000014 radians We can put this into the equation to get r = 1/( 0.0000014) = 711258 AU or 700 thousand times farther away than the sun!!
Stellar distances are so enormous we need special units to describe them. We have already introduced and used the light year. Another unit that is very convenient is the parsec. A parsec is defined to be:
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the
distance at which an object will have a parallactic shift of 1"
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If we use the parsec unit then the distance formula is as simple as it can possibly be: in words:
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distance in parsecs = 1/parallax in
seconds of arc
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If we use this approach then it is easy to tell the distance to 61 Cygni. r = 1/p= 1/0.29 = 3.45 parsecs
The following formula combines absolute and apparent magnitudes with distance measured in parsecs:
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M = m - 5 log(r/10)
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where:
This formula is often re-arranged to read:
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m - M = 5 log(r/10)
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and the term m - M is called the distance
modulus.
We can use a graphical approach to the distance modulus equation:
| for distances less than 100 pc | click here, |
| for distances between 100 pc and 1000 pc | click here. |
Example:
To see how to use this relation let's consider a star that we know (from
parallax measurement p = 0.02") to be 50 pcs or about 150 ly away. If the
star appears to be 3rd magnitude what is its absolute magnitude?
Solution:
Click on the graph for r < 100 pcs. The red line shows this case. When
r = 50 pcs the distance modulus is 3.5. This means that m-M = 3.5. Re-arranging
yields: M = m - 3.5. Since the star has an apparent magnitude of 3, M = 3
-
3.5 = -0.5.
Distance Modulus Calculators:
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Time to take a Quiz! |
Seeds: Chp
9; 172-178
Kaufmann: Chp19;
336-349
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