## How do astronomers determine the size and distances of stars?

To get distances, we use a variety of techniques. The most basic one is geometric parallax. By photographing the same star 6 months apart from points 1 and 2 in earth's orbit, the shift of the star relative to more distant background stars when R = 1 Astronomical Unit amounts to 1 second of arc at 1 parsec ( 3.26 light years), 1/2 arcsecond at 2 parsecs, 1/10 arcsecond at 10 parsecs etc. By the way, at 1 parsec, an arcsecond also subtends 206265 astronomical units.

The Hipparcos astrometric satellite has determined the distance to over 100 thousand stars in this way. Read an ESA Press Release about the mission accomplishments. For example, the distances to the Nearest 10 stars can be found in their Table of 150 closest stars which I reprint below:

```Name					Parallax
Alpha Centauri C			772.33
Alpha2 Centauri C			742.12
Alpha1 Centauri C			742.12
Barnard's Star			549.01
Alpha Canis Majoris (Sirius)	379.12
Epsilon Eridani			310.75
61 Cygni A				287.13
Alpha Canis Minoris		285.93
61 Cygni B				285.42
Epsilon Indi			275.76
Tau Ceti				274.17

```
Note: the Parallax is measured in 1/arcseconds. To calculate the distance in parsecs you have to take 1000.0 and divide it by the parallax number in the last column above. For example, Alpha Centauri C (Proxima) is at a distance of 1000.0/772.33 = 1.295 parsecs which equals 1.295 x 3.26 = 4.22 light years. Alpha Centauri is at 1000/742 = 1.34 parsecs or 4.39 light years. I leave it as a simple calculator exercise for you to convert the parallaxes above into light years!

The result is a detailed map of the stars near the sun out to many parsecs like the figure below.

Stellar diameters can be measured for some nearby giant and supergiant stars by using a technique called stellar interferometry. The Navy Prototype Optical Interferometer has been operating for over a decade at Mount Wilson Observatory, and routinely measures the angular diameters of bright stars to fractions of a milli arcsecond (0.001 arcseconds) accuracy. The table below shows only a few stars that have had their diameters measured. Once their distances are accurately known...from the Hipparcos Survey...their linear diameters in millions of kilometers can easily be found.

The table below shows the sizes in multiples of the solar diameter for some typical stars that have measured angular diameters in column 5 given in arcseconds. The highest resolution of the Hubble Space Telescope is about 0.046 arcseconds. So it is just able to see Betelgeuse as a resolved 'disk'

```Name              Type      dist.    diameter     Size
Alpha Arietis     K2III    65.9 ly   0.00699      14.8
Alpha Cassiopeia  K0III    150.0     0.00569      27.4
Alpha Persei      F5Ib     592       0.00313      59.3
Alpha Leporis     F0Ib    1280       0.00177      72.5
Betelgeuse        M1Ib     425       0.054       734.4
Antares           M1Ib     520       0.041       682.2
Proxima Centauri  dM5      4.2       0.007         1.0
Polaris           F7 Ib    430       0.00328      45.1
```
The size in kilometers = 3 x 10^13 (d /3.26) (D/3600)/57.3 or 44.6 million x d x D where d = the distance in light years and D is the angular diameter in arcseconds. In terms of solar diameters (1,390,000 km) you get Size = 32 d x D solar diameters. The later formula gives you the above entries in the last column. The super giant star Betelgeuse is 734.4 times the diameter of the Sun.

Here is an image of Betelgeuse showing its actual surface. Note it is not round. Astronomers think this may be due to a large sunspot which dims part of the disk making the shape look irregular.

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