Distance modulus

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The distance modulus is a way of expressing distances that is often used in astronomy.

Contents

[edit] Definition

The distance modulus μ = m − M is the difference between the apparent magnitude m and the absolute magnitude M of an astronomical object. It is derived from the definition of magnitude as the logarithm of the ratio of observed fluxes of astronomical objects:

m1 − m2 = − 2.5log10(F1 / F2)

The observed brightness of a light source is related to its distance by the inverse square law - a source twice as far away appears one quarter as bright. For a single object or two objects of the same luminosity, (F1 / F2) can be replaced by (d2 / d1)2 since

F_1/F_2 = \left(\frac{L}{4{\pi}d_1^2}\right)\left(\frac{4{\pi}d_2^2}{L}\right)

Absolute magnitude is defined as the apparent magnitude of an object when seen at a distance of 10 parsecs, and so the magnitude equation can be written as:

m − M = 5log10(d / 10pc)

Rearranging the logarithms gives

m − M = − 5 + 5log10d

Then, given a distance modulus μ = m − M, the distance in parsecs is given by

d = 100.2(m − M + 5) = 100.2μ + 1

The uncertainty in the distance in parsecs (δd) can be computed from the uncertainty in the distance modulus (δμ) using

\delta d = 0.2 ln(10) 10^{0.2\mu+1} \delta\mu = 0.461 d \ \delta\mu

which is derived using standard error analysis[1].

[edit] Different kinds of distance moduli

Distance is not the only quantity relevant in determining the difference between absolute and apparent magnitude. Absorption is another important factor and it may even be a dominant one in particular cases (e. g. in the direction of the galactic center).

Thus a distinction is made between distance moduli uncorrected for interstellar absorption (whose values would overestimate the distance if used naively) and absorption-corrected moduli.

The first ones are termed visual distance moduli and are denoted by (m − M)v while the second ones are called true distance moduli and denoted by (m − M)0.

Visual distance moduli are computed by calculating the difference between the observed apparent magnitude and some theoretical estimate of the absolute magnitude. True distance moduli require a further theoretical step, that is the estimation of the interstellar absorption coefficient.

[edit] Usage

Distance moduli are most commonly used when expressing the distance to other galaxies in the relatively nearby universe. For example, the Large Magellanic Cloud is at a distance modulus of 18.5[2], the Andromeda Galaxy's distance modulus is 24.4[3], and the galaxy NGC 4548 in the Virgo Cluster has a DM of 31.0[4]. In the case of the LMC, this means that the supernova SN1987A, with a peak apparent magnitude of 2.8, had an absolute magnitude of -15.7, which is low by supernova standards.

The distance modulus is also a preferred way of expressing distances for many observers since the practical effect of increasing distance is to make things appear fainter; eg., a solar type star (M=5) in a galaxy in the Virgo Cluster would have an apparent magnitude (m) of 36, a calculation which can be done quickly in ones head. Since it is apparent magnitudes which are actually measured at a telescope, this way of looking at things serves to highlight the fact that many discussions about distances in astronomy are really discussions about the putative or derived absolute magnitudes of the distant objects being observed.

[edit] References

  • Zeilik, Gregory and Smith, Introductory Astronomy and Astrophysics (1992, Thomson Learning)
  1. ^ J. R. Taylor (1982). An introduction to Error Analysis. Mill Valley, California: University Science Books. ISBN 0-935702-07-5. 
  2. ^ D. R. Alvez (2004). "A review of the distance and structure of the Large Magellanic Cloud" (abstract) 48: 659-665. doi:10.1016/j.newar.2004.03.001. 
  3. ^ I. Ribas, C. Jordi, F. Vilardell, E. L. Fitzpatrick, R. W. Hilditch, E. F. Guinan (2005). "First Determination of the Distance and Fundamental Properties of an Eclipsing Binary in the Andromeda Galaxy" (abstract) 635: L37-L40. doi:10.1086/499161. 
  4. ^ J. A. Graham, L. Ferrarese, W. L. Freedman, R. C. Kennicutt Jr., J. R. Mould, A. Saha, P. B. Stetson, B. F. Madore, F. Bresolin, H. C. Ford, B. K. Gibson, M. Han, J. G. Hoessel, J. Huchra, S. M. Hughes, G. D. Illingworth, D. D. Kelson, L. Macri, R. Phelps, S. Sakai, N. A. Silbermann, A. Turner (1999). "The Hubble Space Telescope Key Project on the Extragalactic Distance Scale. XX. The Discovery of Cepheids in the Virgo Cluster Galaxy NGC 4548" (abstract) 516: 626-646. doi:10.1086/307151.