How do i figure it out: how far is a star with an appearent magnitude of +2 and an absolute magnitude of +4?

Question

Is there a formula for that?

I think the answer is 12.7 light years but i don't know how to get that answer.
thanks!

Answer 1

The absolute Magnitude (we always write it with a capital M to remind us not to get mixed up), is the magnitude the star would appear to be if it were placed at 10 parsecs (32.6163 light-years) from us.

The apparent magnitude indicates how brilliant the star appears at its true distance.

The intensity of light decreases as the square of the distance (push a star three times further, it will appear nine times fainter).

The magnitude scale is geometric. One step in the magnitude sacel corresponds to a ratio of approx. 2.512 (it is the fifth root of 100).
A star of m= 2 is 2.512 times brighter than a star of m=3.
A star of m= 2 is 6.310 times brighter than a star of m=4
...
A star of m= 2 is 100 times brighter than a star of m=7.

Here, we have M=+4 at 10 parsecs and m=+2 at distance X.
+2 is brighter than +4 (the magnitude scale runs backwards with bigger number meaning fainter stars).
+2 is 6.310 times brighter than +4.
Distance X must be 10 parsecs, divided by the square root of 6.310.
X = 10/SQRT(6.310) = 10/2.512 = 3.98 parsecs.
3.98 parsecs = 13 light years.

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The formula to calculate distance based on magnitudes is called the "distance modulus" and it involves logarithm. This is good because powers and roots in normal calculations become simple multiplications and divisions in logarithmic calculations.

m − M = 5*log_10(d / 10pc)

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pc means parsec, a distance unit based on the distance at which the parallax (based on the orbital radius of Earth's orbit around the Sun) is exactly 1 second of angle.

log_10 means; logarithm in base 10 (a.k.a. common logs, not "natural" logs).

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M = +4, m = +2

m − M = 5*log_10(d / 10pc)

2 - 4 = 5*log_10(d / 10 pc)
-2/5 = log_10(d / 10 pc)

to get rid of the log_10, make both sides the power of 10.
(This operation is called: taking the "antilog") On many calculators, this is done by pressing "inverse" then "log". It is also the same as the key marked 10^x.

If a = b, then 10^a = 10^b

Takint the antilog of a log cancels the log and leaves only what is in the bracket:

10^[log_10(x)] = x
antilog[log_10(x)] = x

so, we now have:
-2/5 = log_10(d / 10 pc)

10^(-2/5) = d/10 pc
1/10^(2/5) = d/10

10^(2/5) is the same as (10^2)^(1/5) which is the say to write: the fifth root of 100. That is 2.512 (approximately)

1/2.512 = d/10
10/2.512 = d = 3.981 parsecs

1 parsec = 3.261631 light-years
therefore
d = 3.981 * 3.261631 = 12.9848 light-years.

13 light years
It would be bold to pretend to a lot of accuracy, unless you had determined that m really is +2.0000 and M = +4.0000 (and not +4.0027). Magnitude tables are rarely that precise.