Once these two measures of brightness have been determined the distance to the star can be calculated by the formula:

d = 10

^{(m-M+5)/5}

where d = distance (in parsecs), m = apparent magnitude and M = absolute magnitude.

Here are some sample calculations:

Proxima Centauri

m = 11.01, M = 15.53

d= 10

^{(11.01 -15.53 + 5) / 5 }

= 1.247 parsecs

The distance determined from parallax is 1.3 parsecs so the Distance Modulus calculation is fairly accurate.

The source for the parallax distance is Research Consortium on Nearby Stars (RECONS).

Parallax = 0.76887 therefore distance = 1 / 0.76887 = 1.3 parsecs

Sirius

m = -1.47, M = 1.48

d = 10

^{(-1.47 - 1.48 + 5) /5}

= 2.57

Paralax distance is 2.63 parsecs (

^{1}/

_{0.38002}) which is

also reasonably accurate.

This is all well and good, but how was the Distance Modulus formula determined?

The rest of the post attempts to answer this question.

Warning: the following is fairly mathematical!

*Relating Flux to Distance*

Flux is the energy (light) passing through an area in an amount of time

It is defined as:

F = L / (4πD

^{2})

Note that 4πD

^{2}is the surface area of a sphere.

The observed brightness of a light source is related to its distance by the inverse square law - a source twice as far away appears to be on quarter as bright. For a single object or two objects of the same luminosity:

The L terms cancel as do the 4π terms

Therefore F

_{1}/ F

_{2}= (D

_{2}/ D

_{1})

^{2}- Equation 1

*Converting Luminosity to Magnitude*

It is important to realise that luminosity is linear while magnitude is logarithmic.

A difference of 5 magnitudes corresponds to a ratio of 100 in luminosity.

L

_{1}/ L

_{2}= x

^{ΔM }

*ΔM is a change in M*

100 = x

^{5}

*5 M difference = 100 L difference*

2.511887

^{5}= 100.0001132

It is conventional to abbreviate this to 2.5

^{5}which equals 97.66 .

Therefore:

L

_{1}/ L

_{2}= 2.5

^{ΔM }- Equation 2

*Relating Luminosity to Distance*

We need to find the ratio of luminosity required to produce the same flux from different differences.

F = L / (4πD

^{2})

L = F4πD

^{2}

L

_{1}= F4πD

_{1}

^{2}

L

_{2}= F4πD

_{2}

^{2}

Note: there is no F

_{1}or F

_{2}as we are looking for luminosity

required to produce the same flux from different differences, ie F = F

L

_{1}/ L

_{2}= F4πD

_{1}

^{2}/ F4πD

_{2}

^{2}

= D

_{1}

^{2}/ D

_{2}

^{2}

= (D

_{1}/ D

_{2})

^{ 2 }- Equation 3

Note this is different to the flux / distance relationship.

*Relating Magnitude to Distance*

As:

L

_{1}/ L

_{2}= 2.5

^{ΔM}

and

L

_{1}/ L

_{2}= (D

_{1}/ D

_{2})

^{ 2 }

therefore

2.5

^{ΔM}= (D

_{1}/ D

_{2})

^{ 2 }

ΔM = 2.5log

_{10}(D

_{1}/ D

_{2})

^{ 2 }

= 5log

_{10}(D

_{1}/ D

_{2})

Note that ΔM = M

_{2}- M

_{1}therefore

M

_{2}- M

_{1}= 5log

_{10}(D

_{1}/ D

_{2})

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

m - M = 5log

_{10}(D/10)

= 5(log

_{10}D - log

_{10}10)

= 5(log

_{10}D - 1)

= 5log

_{10}D -5

*Solving for Distance*

if m - M = 5log

_{10}D -5 then

m - M + 5 = 5log

_{10}D

(m - M + 5)/5 = log

_{10}D

Therefore D = 10

^{(m - M + 5)/5}

which is the relationship that we set out to prove!

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