## Friday, July 22, 2011

### Solving Simple Astronomy Problems (4)

We pick up where we left off, with the problems then the solutions, for part (4):

Problems:

1) A star with an apparent magnitude of +1.0 is located at a distance of 40 pc. What is its absolute magnitude M? What is its distance modulus?

(2) Find the apparent magnitude of the star Vega if its absolute magnitude is +0.3 and its parallax angle p = 0."123.

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

(4)Altair in the constellation Cygnus is 16.4 LY distant with an apparent magnitude m = +0.8 and absolute magnitude M = +2.3. Verify its absolute magnitude using the inverse square law for light.

Solutions:

1) We know m = +1, and D = 40 pc.

Then, from the distance modulus, the absolute magnitude M:

M = m + 5 - 5 log D

M = +1 + 5 - 5 log 40

or: M = +6 - 5 log 40

log 40 = 1.602

So:

M = +6 = 5 (1.602) = +6 - 8.01 = -2.01

Then: the distance modulus is:

(m - M) = +1 - (-2.01) = 3.01

2) We have: M = +0.30 and p = 0."0123

The apparent magnitude of Vega may be determined from the distance modulus in the form:

(m - M) = 5 - 5 log p

So: m = M - 5 - 5 log p

or:

m = +0.30 - 5 - 5 log (0.123)

m = -4.70 - 5 (-0.91) = -4.70 + 4.55 = - 0.15

3) Since the trigonometric parallax p = 0."093 we can find the distance in parsecs directly, since:

D = 1/p

Thus, D = 1/ (0.093) = 10.75 pc

But we need D in light years, and know 1 pc = 3.26 Ly, so:

D = 10.75 pc (3.26 pc/Ly) = 35 Ly, approx.

4) Altair has an apparent magnitude m = +0.8 at 16.4 Ly, but 16.4 pc = 5 pc

I.e. 5 pc x 3.26 Ly/pc = 16.4 Ly

Absolute magnitude M is based on a standard distance of 10 pc.

By the inverse square law for light:

(D/D')^2 = (B'/ B)

Where B'/B denotes the brightness ratio.

D/D' = ½

So: B' = (½)^2 B = ¼ B

I.e. the brightness at 10 pc must be decreased by a factor 4.

Then:

(2.512)^n = 4

n log (2.512) = log 4

n (0.4) = 0.60

n = 0.60/0.40 = 1.5

In other words, to get M one must ad (1.5) to m. Hence:

M = (+0.8) + 1.5 = +2.3