Absolute magnitude and the distance modulus

Builds on: Sections 4.1, 4.2 and 1.1
Reading time: 30 min
Exercises: 5
Olympiad problems: 0

Sirius, at magnitude −1.44, is the brightest star in the night sky, and Rigel, the bright blue star at the foot of Orion, is a little fainter at magnitude 0.18. Sirius is only 2.64 parsecs away, however, and Rigel about 265 parsecs, a hundred times further. Moved to the same distance, Rigel would outshine Sirius more than two thousand times over. In this section we put stars at one standard distance to compare the light they really give out, and we find the rule that links how bright a star looks, how bright it is, and how far away it lies.

4.3.1Absolute magnitude

Before you readMake a guess first

A star is moved from 10 parsecs away to 100 parsecs away. By how much does its magnitude change?

The magnitude of a star as we see it, called its apparent magnitude $m$ , mixes two things, how much light the star gives out and how far away it is. To compare the stars themselves we imagine each one moved to a distance of exactly 10 parsecs. The magnitude it would then have is its absolute magnitude $M$ . The distance of 10 pc is only a convention, chosen long ago because it is typical of the nearest stars.

A star at distance $d$ moved to 10 pc would have its flux multiplied by $(d / 10 pc)^{2}$ , by (4.1). By (4.2), its magnitude would fall by $2.5 \log_{10} (d / 10 pc)^{2} = 5 \log_{10} (d / 10 pc)$ , so the difference $m - M$ depends on the distance alone. It is called the distance modulus.

ResultThe distance modulus(4.4)

\begin{aligned} m - M & = 5 \log_{10} \frac{d}{10 pc} \\ = 5 \log_{10} d - 5 \end{aligned}

$m$	the apparent magnitude of the star
$M$	its absolute magnitude, the magnitude it would have at a distance of 10 pc
$d$	its distance, in parsecs in the second line

The rule assumes that no dust or gas between the star and us dims its light. Turned round, it gives the distance as

d = 10^{(m - M + 5) / 5}

pc. A distance modulus of 0 means 10 pc, and every 5 more magnitudes multiply the distance by 10.

For Sirius, $M = - 1.44 - 5 \log_{10} 2.64 + 5 = 1.45$ , and for Rigel, $M = 0.18 - 5 \log_{10} 265 + 5 = - 6.94$ . Rigel is 8.39 magnitudes brighter than Sirius at equal distances, which by (4.2) is a factor of 2270 in the light they give out.

The distance of a nearby star comes from its $p$ , the angle that the radius of Earth's orbit covers as seen from the star (Section 1.1). A star with a parallax of 1″ is 1 parsec away, and since the angle shrinks as the distance grows, a star with parallax $p$ in arcseconds is $1 / p$ parsecs away. The distance modulus can then be written as $m - M = - 5 \log_{10} p - 5$ , with $p$ in arcseconds.

Quick checkThe absolute magnitude of the Sun

The Sun has apparent magnitude −26.74 as seen from Earth, 1 au away. One parsec is 206 265 au. What is the Sun's absolute magnitude?

4.3.2Luminosity from absolute magnitude

Two stars at the same distance have fluxes in the ratio of their luminosities, so a difference in absolute magnitude is a ratio of luminosities, read through (4.2). Measured against the Sun, whose absolute magnitude in the V band is 4.83, it gives the luminosity of any star in units of the Sun's.

ResultLuminosity from absolute magnitude(4.5)

\frac{L}{L_{⊙}} = 10^{- 0.4 (M - M_{⊙})}

$L, L_{⊙}$	the luminosities of the star and of the Sun
$M, M_{⊙}$	their absolute magnitudes in the same band, with $M_{⊙} = 4.83$ in the V band

Used with magnitudes in one band, such as V, the rule compares only the light in that band. To compare the total power over all wavelengths we use bolometric magnitudes, which Section 4.7 introduces.

Example 4.5How luminous is Vega?

Vega has apparent magnitude 0.03 and lies 7.68 pc away. What are its absolute magnitude and its luminosity in the V band, in units of the Sun's?

Solution

By (4.4),

M = 0.03 - 5 \log_{10} 7.68 + 5 = 0.60 .

Then by (4.5),

\frac{L}{L_{⊙}} = 10^{- 0.4 \times (0.60 - 4.83)} = 49.

AnswerVega has absolute magnitude 0.60, and in the V band it gives out about 49 times as much light as the Sun.

4.3.3Dust on the way

Space between the stars holds thin clouds of dust, which absorb and scatter some of the light that passes through them. A star seen through dust looks fainter than its distance alone would make it, by an amount $A$ in magnitudes called the . The distance modulus then gains one term, $m - M = 5 \log_{10} d - 5 + A$ . In the V band, near the plane of our Galaxy, $A$ is roughly 1 magnitude for each kiloparsec of path, though it changes a great deal from one direction to another. Section 4.4 treats the same dimming in Earth's atmosphere, and Section 4.7 shows how the colour of a star reveals how much dust lies in front of it.

Example 4.6The brightest supernova of our timefinish it

Supernova 1987A exploded in the Large Magellanic Cloud, about 50 kpc away, and at its brightest it reached magnitude +3. Ignoring dust, what was its absolute magnitude at its peak?

The start

The distance is $50 000$ pc, so by (4.4) the distance modulus is $5 \log_{10} 50 000 - 5 = 18.49$ .

Your stepNow find

M

Common errorsSection 4.3

The distance in $5 \log_{10} d - 5$ must be in parsecs. A distance in light-years or metres gives a wrong modulus.
A smaller absolute magnitude means a more luminous star, as with apparent magnitudes. Rigel, with $M = - 6.94$ , gives out far more light than Sirius, with $M = 1.45$ .
When dust dims a star and the extinction is left out, the star seems further away than it is, since all of its dimming is put down to distance.

ExercisesSection 4.35 questions

4.3.1
What is the distance modulus of a star 1 kpc away, if there is no dust?
magnitudes
4.3.2
A star has apparent magnitude 8.0 and absolute magnitude 3.0. How far away is it?
pc
4.3.3
A star has parallax 0.25″ and apparent magnitude 5.00. What is its absolute magnitude?
4.3.4
A star has absolute magnitude −0.17 in the V band. How many times as luminous as the Sun is it in that band?
times
4.3.5
Two stars have the same absolute magnitude. One is 10 pc away and the other 1000 pc away, with no dust between. How much fainter does the further star look?