# Light Time, Heavy Math

## © 2001 by Robert C. Moler

Astronomical distances are mind bogglingly huge. Hence the term astronomical number to denote something really big like, say, governmental appropriations.

Distances within the solar system are measured in astronomical units (AU), the mean distance from the earth to the sun. That distance is 149,597,870 kilometers or 92,955,807 miles. Outside the solar system the preferred unit used by astronomers is the parsec. A parsec is the length of the long leg of a right triangle, whose other leg is one astronomical unit. The angle between that long leg and the hypotenuse at the end of this long triangle is one second of arc, or 1/3600th of a degree. A parsec is 30,857,000,000,000 kilometers or 19,174,000,000,000 miles, that's 19 trillion. While these numbers are impressive, they can be more conveniently written is scientific notation. The parsec value in miles can be written as 1.9174 X 1013 in books or 1.9174E+13 in computer printouts.

This method to find the distance of objects using one leg of a triangle as a base is called parallax, Thus the name parsec comes from parallax second. The parsec is a natural unit. calculations of distance even beyond the distances that can be measured by the parallax method still use parsecs.

For most of us, a parsec isn't easy to relate to. Thus the light year was invented. It is a huge unit. It is the distance light travels in one year. Now light travels at the fastest velocity know. It travels at the ultimate speed limit of the universe according to Einstein. This turns out to be 299,792.458 kilometers a second or 186,282.397 miles per second. The number of seconds in a year are 60 seconds times 60 minutes times 24 hours times 365.25 days in a year or 31,557,600 seconds. Multiply light speed in an second by the number of seconds in a year, and we get a light year of 5,878,625,000,000 miles, give or take. We usually round that off to 5.8 or 6 trillion miles. It's the approximate national debt at a dollar a mile. (And they say we're running a surplus.)

The neat thing about the light year is that the distance is directly related to time. When we look out to an object, say ten light years away, we are looking back ten years in time. While the speed of light is a barrier to interstellar travel, it does turn ordinary telescopes into time machines, though ones that can only look back.

The parsec and the light year are comparable units. there's 3.26 light years in a parsec. Now you have some tools to measure the universe. Most catalogs of stars don't state distances in light years or parsecs. rather they have a column headed parallax or simply p, the Greek letter pi. Check the units, but it is usually in seconds of arc. If so, the value will be less than 1. No star, other than the sun, is less than a parsec away. To convert parallax to parsecs, divide 1 by the parallax. To convert parallax to light years divide 3.26 by the parallax. Sirius, according to my Atlas Coeli II Catalog, has a parallax of 0.373 arc seconds. 1/0.373 gives 2.68 parsecs. 3.26/0.373 gives 8.7 light years. By comparison the moon is 1.5 light seconds away, the sun 8 light minutes.

A star's parallax is measured by viewing it at various times of the year, and measure its shift against more distant stars. Until the Hipparcos satellite the limit for these measurements was just over 100 light years. Hipparcos and a new generation of satellites will extend that farther. Parallax is the only direct method of distance measurement that actually doesn't use echo location, like radar or using a tape measure. Beyond that one has to extrapolate.

Here brightness is the key. The closer a star is the brighter it appears. But how bright is the star to begin with? How bright it appears from earth is called apparent magnitude. I'll leave the magnitude scale itself for another time, except to state that the brighter the star the lower the magnitude. For the nearest stars, whose distance is known by parallax, we can calculate its brightness for some standard distance. Astronomers have chosen the distance of 10 parsecs. A star's brightness adjusted for a distance of 10 parsecs is called the star's absolute magnitude. The formula m - M = 5 log d - 5 expresses the relationship of M, absolute magnitude; m, apparent magnitude, and d, distance in parsecs. We can calculate a star's absolute magnitude by making d equal to the star's distance in parsecs, 2.68. Log d (the common logarithm of d) then becomes 0.43 Sirius' apparent magnitude is -1.37. So substituting m and log d we have -1.37 - M = 2.15 - 5 + 1.37. Solving for M we have: -M = 2.15 - 5 + 1.37. So M = +1.5, which is Sirius' absolute magnitude. The sun by comparison is +4.5, a much dimmer star.

In order to use the magnitude difference to determine the distance of a star, you have to know it's absolute magnitude. That's tough to do without knowing its distance. Beyond the distance where parallax can be used it's best to work with clusters of stars. Here we have a broad range of stars. There is one star cluster within the range of the parallax method, barely. It is the Hyades, the stars of the face of Taurus the Bull, at 151 light years.. In making a graph of a star's brightness versus its surface brightness or color, Astronomers Hertzsprung and Russell found there was an order to where the stars were plotted. There was a narrow band of stars that charted in a diagonal line from cool and dim to hot and bright. This band is known as the main sequence. If we compare the difference in brightness of main sequence stars of the distant cluster with those of the Hyades. The m-M formula above works here too.

All the tricks astronomers have learned to discern distances are based on these principles.

Questions? Send Email to me at bob@bjmoler.org