"Math Drives The Universe?" Of course! (First of 3 Parts)

The ASTRONOMY magazine article entitled 'Speaking the Language of the Universe' (December, p. 24) is one every math phobe needs to read- especially people who also appreciate the wonders of the night sky.

But sadly, as the piece by Bob Berman observes, while "professional astronomers use math all the time" - see e.g. some of the (free)  journal articles here:

http://iopscience.iop.org/0004-637X/743/2

Hobby astronomers generally "don't have math in their bones".  As Berman adds:

"They hate the subject. Perhaps it reminds them of school. Square roots and standard deviations bog down articles for them, and science writers oblige by leaving out equations altogether."

Which is sad beyond measure, because indeed, as Berman goes on to show, math is the language of the universe. Sure it's easy to simply stand beneath the stars and simply gaze at galaxies, planets and the Moon through a telescope - Oooing and ahhing all the time. But while wonder and awe can be the keynote at that level, it can't lead to any understanding of what those objects actually are, what they're made of, how they move or how they might affect our planet. For that,  deeper analysis is needed, i.e. into the relationships between physical parameters that define them.

Just to know how distant an object is - say a nearby star- depends on math, and it is critical given that the distance can allow other properties to be determined. Consider the famous "parallax method" to obtain the distance to nearby stars. This method  can apply to all stars within a distance of maybe 50 parsecs (1 pc = 3.26 Ly) or those for which a measurable parallax angle p exists.  The geometry shown below is useful to this end:

The angle p is obtained by taking photographs of the same star six months apart (i.e. from opposite sides of Earth's orbit) and comparing the two positions. One can thereby obtain the distance, D from:

D = r/ tan (p)

The relationship is such that for p = 1 arcsec the distance of the star would be 1 parsec (e.g. par-allax sec-ond). an angle of 1 arcsec = 1" = 1/3600 degree. So we see it is an extremely tiny angle. similarly, if the angle p = 1/10" then D = 10 parsecs, so we perceive a reciprocal relationship such that D = 1/p", though we must ensure the units are consistent.

In many applications, such as the budding undergrad student meets in 1st year Astronomy, the parallax angle p is merged with the equation for the "distance modulus" -  which makes use of the absolute magnitude M (see previous blog on this) and apparent magnitude m. In this way, estimates of the star's energy output, and brightness can be made.

Then, if D is the distance, the usual expression for distance modulus is:

(m - M) = 5 log (D/10) = 5 log D - 5 log 10 = 5 log D - 5

But: D = 1/p
so:

(m - M) = 5 log (1/p) - 5

Or: (m - M) + 5 = 5 log p

A sample problem, such as an Astronomy 201-202 student would have to do for homework, is as follows:

Barnard's star has an absolute magnitude of +13.2 and an apparent magnitude m = +9.5. Find its distance in LIGHT YEARS.

The solution is based on using the parsec form of the distance modulus:

Then:

(m - M) = 5 log (1/p) - 5

(9.5 - 13.2) = 5 log(1/p) - 5

-3.7 = 5 log (1/p) - 5

5 log (1/p) = (5 - 3.7) = 1.3

log (1/p) = (1.3)/5 = 0.26

Taking anti-logs:

1/p = D = 1.81 pc

But 1 pc = 3.26 Ly, so D = (1.81 pc)(3.26 Ly/pc) = 5. 9 LY

A problem to try on your own:

The star Pollux in the constellation Gemini has a parallax angle p = 0."093. Find its distance in light years.

(For those who need a refresher on stellar magnitudes go to:  http://brane-space.blogspot.com/2011/07/tackling-simple-astronomy-problems-3.html

Next: Math examples that Astronomy majoring sophomores might confront.