Absolute magnitude, M

1. Parsec and light year:

  • ⇒Light year:

  • A light year is the distance travelled by light in one year. So
  • [math]1 \text{ light year} = \text{speed of light} \times \text{number of seconds in 1 year} \\
    1 \text{ light year} = 3 \times 10^8 \, \text{m.s}^{-1} \times 3.155 \times 10^7 \, \text{s} \\
    1 \text{ light year} = 9.46 \times 10^{15} \, \text{m} [/math]


    Figure 1

  • Some examples of average distances in light years are:
  • The distance to the star Sirius from the sun is 8.6 light year.
  • The distance to the Andromeda galaxy from the sun is 2.5 million light years.

  • ⇒ Parsec (pc):

  • Nearby objects appear to move relative to far-away objects, when viewed from a different angle.
  • Some stars are closer to us than others because they appear to move slightly as we view them at different times of year. Figure 3 (not drawn to scale) shows the idea.
  • In January, for example, we look at a nearby star, then six months later we look at it again. The star appears to have moved relative to more distant stars, which are very far away.
  • The angle shown in the diagram is called the parallax angle. Because even these ‘nearby’ stars are actually several light years away, this parallax angle is very small.
  • We can calculate the distance from the Earth to a star.
  • [math] tan⁡θ = \frac{1AU}{d} [/math]

  • Or because θ is very small:

  • [math] ⁡θ = \frac{1AU}{d} \\ d = \frac{1AU}{θ} [/math] 

  • Remember that  must be measured in radians. This relationship leads to a new measure of distance, which is directly related to the angle θ. When θ is 1 second of arc, we say that the distance is 1 parsec.

  • [math] \text{1 second of arc} = \frac{1}{3600} degree = 4.85 \times 10^{-6} rad [/math]
  • Therefore
  • [math] 1 \text{ parsec} = \frac{1 \, \text{AU}}{4.85 \times 10^{-6}} \\
    1 \text{ parsec} = \frac{1.5 \times 10^{11}}{4.85 \times 10^{-6}} \, \text{m} \\
    1 \text{ parsec} = 3.09 \times 10^{16} \, \text{m} \\
    1 \text{ parsec} = 3.26 \, \text{light years} [/math]
  • If the measured parallax angle is smaller, then the distance to the star is further. The distances to galaxies are expressed are often expressed in megaparsec (Mpc).

2. Absolute magnitude:

  • Figure 4 shows light spreading out from a light source.


    Figure 2 light spreading out from a
    light source

  • The light travels further from a source, it spreads over a larger area, so its intensity decreases.
  • When the distance from the source doubles, the intensity of the light reduces by a factor of 4, because the light spreads over four times the area. This is called the inverse square law for intensity:
  • [math] I ∝ \frac{1}{d^2} [/math]
  • This idea is important when it comes to comparing the brightness of stars.
  • Earlier you met the idea of apparent magnitude – these measures how bright a star appears to be.
  • However, stars appear brighter if they are close to us.
  • So, to compare the brightness of two stars, we need to consider how bright they would appear to be if they were exactly the same distance from us.
  • The distance that is chosen for comparison is 10 parsecs.
  • A star’s absolute magnitude is the apparent magnitude it would have if it were placed 10 parsecs away from us.
  • In applying the inverse square law for stars, we assume that no light is absorbed by interstellar material such as gas or dust.
  • The apparent magnitude, m, and the absolute magnitude, M are linked by the following formula, in which d is the distance of the star from us, measured in parsecs:
  • [math] m -M = 5 log_{10} (\frac{d}{10}) [/math]
  • This formula combines the idea of the inverse square law for light and the standard reference distance of 10 pc.
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