Star Distances
168-170
Ha! The heliocentric model is WRONG!!! It predicts that the earth should move and so we would see the stars from different locations during the year and thus we should see the closer stars shift back and forth. Try this with your finger. Hold your hand at arms length and look at your index finger, first with your right eye and then with your left eye. As you blink your eyes you should see your finger shift back and forth in relation to background objects. This is parallax. The stars don't seem to move like this. No parallax - no heliocentrism.
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) actually does shift 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 a large classroom or a penny seen from a distance of 10.7 km!! Despite this tiny amount, the measurement of stellar parallax was dramatic. Struve was awarded the Royal Astronomical Society's Gold Medal in 1841 and Sir John Herschel (son of Sir William Herschel) 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. Figure 8.1 provides an animation (and is highly exagerrated!) See if you can spot the two stars displaying parallactic shifts. This is a "bogus" animation - no stars show this large a shift as seen from our solar system. Also - the entire process of shifting back and forth takes a year!
| Figure 8.1 Animation of highly exaggerated parallactic shift. |
Figure 8.2 provides a schematic representation of parallax and introduces the parallactic shift as an angular measure of the amount of shift an object undergoes.
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| Figure 8.2 How to mesure the parallax or parallactic shift of an object. |
"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:
where:
- d = distance to the star expressed in Astronomical Units,
- p = the parallax angle (measured in seconds of arc)
Example 8.1 An astronomer observes a distant object and detects a parallactic shift of 1.2". How far is this object from the sun?
Solution: This is straight forward - just use and insert appropriate values to get 
.
The object is located approximately 170 thousand astronomical units from the sun.
Example 8.2 The star 61 Cygni has a parallactic shift of 0.29". How far away is it?
Solution: Just use . We can put this into the equation to get d= 711258 AU or 700 thousand times farther away than the sun!
The Parsec - an easier way!
As you saw in the previous two examples, astronomical distances are very large and the numbers quickly become too unwieldy to use. To avoid this problem astronomers have adopted a different unit for measuring stellar distances. If you rewrite the distance equation as 
and then set the distance "d" at 206,265 AU you can see that p = 1". This means that an object located 206,265 AU away will undergo a parallactic shift of 1" . By adopting the new unit of distance called the parsec where 1 parsec = 206,265 AU we get a much simpler distance formula:
the distance at which an object will have a parallactic shift of 1" |
If we use the parsec unit then the distance formula is as simple as it can possibly be: in words:
distance in parsecs = 1/parallax in seconds of arc |
Aaargh... Why So Many Units!?
One size rarely fits all! This is also true in astronomy. Scientists adopt units that make mathematical formula as simple as possible. So far you have used familiar units like meters and kilometers and but have also been introduced to the light year, the astronomical unit and now the parsec. This is really no different than descriing time in seconds, hours, days or years. You easily navigate these kinds of units and with pracice you will do the same with the units used in astronomy. The following table will help:
| 1 AU | 1.496 x 1011 m |
| 1 pc | 206,265 AU |
| 1 pc | 3.086 x 1016 m |
| 1 light year (ly) | 9.461 x 1015 m |
| 1 pc | 3.262 ly |
| Table 8.1 Conversion Factors | |
Example 8.3 The star 61 Cygni has a parallactic shift of 0.29". How far away is it in parsec units?
Solution: Simple! Using 
just plugin the numbers to get d = 1/0.29 = 3.45 pc.
Example 8.4 What is the distance to 61 Cygni expressed in light years?
Solution: Use the Table 8.1 to convert between pc and ly. 1 pc = 3.262 ly, so 3.45 pc X (3.262 ly/pc) = 11.25 ly. 61 Cygni is a little over 11 light years away.
Illustrating Parallax
Figure 8.3 provides you with an 
applet that allows you to simulate parallax. On the far right of the applet is a group of stars so distant that they show no detectable parallax. These are the "background" stars. By changing the position of the earth (the buttons on the applet) or by dragging the "target star" with the mouse you can begin to develop an understanding of how parallax works. Try it out!
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| Figure 8.3 Applet depicting parallax and parallactic shift. |
Parallax - Then and Now
Prior to the advent of space based observatories parallax measurements could only be made for a small number of stars (a few hundred). Turbulence in the atmosphere hampers our ability to measure the small parallactic shifts . In practice the smallest parallactic shift measurable from Earth is about 0.02" (with a precision of 10%). However, as you will see in future sections, distance is one of the most fundamental properties that we need to know in order to understand the nature of stars. For this reason in 1989 the Hipparcos space satellite was launched with the goal of measuring parallax to a precision of +/- 0.001" .
The Hipparcos satellite was a success and has increased dramatically the number of stars with precise parallax measurements. Hipparchus has provided precise distance determinations for about 120 000 stars. Of this number, about 20 000 stars have parallaxes known to an accuracy better than 10%.
Hipparchus operated from the fall of 1989 to the summer of 1993.
Practice: Try the following questions about parallax. You may find the applet shown in Figure 8.3 helpful in understanding parallax and answering these questions.
- Star "A" is twice as far away as star "B". How would the parallaxes of these two stars compare? Try showing this by using the applet given above.
- How far away is Capella ( p = 0.080")?
- What is the parallax for Aldebaran which is 60 ly away? (Hint: make sure to convert the distance to pcs!)
- The Hipparchos satellite telescope was tiny - even by amateur standards . The primary mirror of the telescope measured a mere 29 cm across. Despite this Hipparchos produced data far superior to that possible with Earth based telescopes more than 10 times as large. Why was Hipparchos so much better at measuring parallax than ground-based telescopes?
To understand how Astronomers are able to determine the distances to the stars
Chapter 8.1
Stellar Parallax is the angular shift that a star exhibits when viewed from different locations in the Earth's orbit.
paralactic shift is the apparent motion of a star that is showing the effects of parallax.
The second of arc unit is commonly used in astronomy. One degree of angle measure can be divided into 60 minutes of arc - each of which can be divided in 60 seconds. Therefore, one degree consists of 60 x 60 = 3600 seconds of arc. Or .. 1 second of arc is 1/3600 th of a degree. The notation " is used to denote seconds of arc.
One parsec is the distance at which an object will have a parallax of 1"