The Dimensionality of the Cosmological Distance Modulus Equation

 

The Dimensionality of the Cosmological Distance Modulus Equation

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

Astronomers generally use the magnitude system to measure a star’s brightness. The apparent magnitude of a star, m, is its apparent brightness. Absolute magnitude is the apparent magnitude a star would have if it were located at a distance of 10 parsecs. (pc) = 32.6 ly away from Earth. The absolute magnitude of a star M[1] is a measure of the intrinsic brightness of the star while the apparent magnitude (brightness as observed from Earth) m depends on both the intrinsic brightness and the distance to the star. The difference between the two, m – M, depends only on the distance. This quantity called the distance modulus, is related, under ideal (no “stardust”) occurrence, to the distance d by

m – M = 5logd – 5    … (1).

where d is in pc.

 

If the distance modulus is negative, the star is closer than 10 pc, and its apparent magnitude is brighter than its absolute magnitude. If the distance modulus is positive, the star is farther than 10 pc and its apparent magnitude is less bright than its absolute magnitude. For example[1], Sirius has m – M = – 2.85. This value is negative and Sirius is closer than 10 pc. Betelgeuse has m – M = 5.59. This value is positive and Betelgeuse is more than 10 pc distance.

The absolute magnitude M of a star is expressed as its apparent magnitude if placed at a distance of 10 pc. So, if we can measure the apparent magnitude of a star m and calculate its absolute magnitude M and the distance d to the star. Similarly, if we know the distance to a star d_m[2], measured by parallax (or some other methods), it is easy to calculate the absolute magnitude of a star M from its apparent magnitude m.

For convenience, we write eqn. (1) in the log form

log d = 1/5(m – M) + 1   … (2).

After the conversion of this equation into the exponential form we have

d = 10^{1/5(m – M) + 1} … (3).

The dimension[1] of the left side of this expression is L but its right side is dimensionless. Hence, eqn. (3) is dimensionally incorrect. The simplest way to make this equation dimensionally correct is to multiply its right side with the L dimensionality constant δ equals or close to 1 and expressed in length unit of d (e.g. parsecs, light years, meters, etc)

d = 10^{1/5(m – M) + 1}δ    … (4).

Since logδ is equal or close to 0 the log form of this equation is identical to the log form of distance modulus eqn. (2).

 

Let us consider the star Rigel[1] a star in the constellation of Orion (Fig. 1).  Using the most recent figures given by the 2007 Hipparcos data (based on the parallax to Rigel), this star is about 265 pc (865 ly) away from Earth. Rigel has an apparent magnitude of m = + 0.18[2] and an absolute magnitude of M = – 6.7.

 

To estimate the distance d to Rigel, we must first calculate the distance modulus: m – M = 0.18 – (– 6.7) = 6.88.   This indicates that Rigel is more than 10 pc away. Then we find that (m – M)/5 = 6.88/5 + 1 = 2.376. Finally, we calculate d = 238 pc (775 ly).

Fig. 1. The Orion Constellation

The difference between тхе measured distance d_m and тхе calculated distance d is about 10 %. This difference is due to the distance modulus error as well as to the L dimensionality constant δ. 

Cepheid variables method is one of the most direct ways to measure distance d_m from about 1 kpc to 50 Mpc. The absolute magnitude M and apparent magnitude m of a Cepheid star can be also related by the distance modulus equation (1), and its distance d can be calculated. If in a particular case, there is a difference between тхе measured distance d_m and тхе calculated distance d this can be also partly attributed to the L dimensionality constant δ, as noted above.

Type Ia supernovae are all caused by exploding white dwarfs which have companion stars. Like Cepheid variables, they can be used as standard candles. Type Ia supernovae can be used to measure distance d_m from about 1 Mpc to over 1000 Mpc. Astronomers can measure the apparent magnitude of a supernova, knowing what its absolute magnitude is. They can then use the distance modulus equation (1) to calculate the distance d to the supernova, and the galaxy that it is in. A type supernova Ia, SN 2011fe, exploded in the galaxy NGC 5457 {1}. Its apparent magnitude m = 10 and a typical Ia-type supernova has an absolute magnitude M = – 19.3.  Using the formula (1), we calculate logd = 1/5(m – M) + 1 = 1/5(10 + 19.3) + 1 = 6.86. Then, we found that d = 7.25 Mpc (23.6 Mly). The currently accepted distance of NGC 5457, determined by the Hubble Cepheid variable measurements, is about 6.75 Mpc. The difference between the measured d_m and the calculated d is about 7.5 %. This difference can be also attributed to the distance modulus error as well as to the L dimensionality constant δ.

Reference

{1} D. Eagle, From Casual Stargazer to Amateur Astronomer. Springer (2014).

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[1] In fact, the absolute magnitude “M” of a star is expressed as its apparent magnitude if placed at a distance of 10 pc.

[2] All data is from the Australia Telescope National Facility.

[3] The subscript m stays for measured.

[4] Just to remind the reader that M, L and T are the symbols for basic dimensions: mass, length and time, respectively. 

[5] All data is from the Universe Guide.

[6] Rigel has an apparent magnitude ranging from 0.05 to 0.18.