Tuesday, September 20, 2022

Looking At Stellar Luminosity & Color Index For Main Sequence Stars

Understanding the importance of stellar luminosity or a star's intrinsic brightness is related to its mass,  It is important to note here that the Mass-Luminosity relation only applies to stars on the Main Sequence, hence gives the mass in terms of the solar mass (the Sun being used as a 'standard' for the stars on the Main Sequence).

From the Mass-Luminosity relation we have the basic relation:

L’/L = (M’/M)3.5

Or Log (L’/L) = 3.5 Log (M’/M)

where L, M refer to solar values and L', M' to stellar values.

Example 1:  For Sirius A,

M' = 2.13 solar masses, so (M’/M) = 2.13, and:

3.5 (Log 2.13) = Log (L'/L) = 3.5 (0.3283) = 1.149

But: antilog (1.149) = 14.09

Or: L' = 14 L approx.

Hence, Sirius A is about 14 times more luminous than the Sun.

Example 2:

The intrinsic brightness (luminosity) of Regulus exceeds the Sun's by a factor 120. Find the approximate mass of Regulus.

Here: L'/L = 120 so Log (120) = 3.5 Log (M'/ M)

and we are seeking to find M' in terms of M.

Log (120) = 2.079 = 3.5 Log (M'/ M)

Or:

0.594 = Log (M'/M)

Taking antilogs of each side:

3.93 = (M'/M) or M' = 3.93 M

Therefore Regulus is approximately 4 times the mass of the Sun.

We already saw the use of simple apparent and absolute magnitudes, e.g.

But in stellar properties' analysis we need to refine this to deal with "absolute bolometric magnitudes" because the brighter stars (or spectral class O and A mainly) require special "color" corrections usually referred to as "bolometric corrections". This refined system of "bolometric magnitudes" is thereby adjusted so the bolometric corrections are small for stars like the Sun (e.g. G class or later) but large for very hot stars where most of the radiated energy is in the unobservable ultraviolet (UV). The Table below gives bolometric corrections for different temperatures and spectral types.

Note the difference (B - V) is the "color index" representing the difference in magnitudes mB and mV, e.g. (mB - mV) where mB is the apparent magnitude from a blue filter and mV is the apparent magnitude from the standard yellow or visual filter - most sensitive to wavelengths near 550 nm.

If the absolute bolometric magnitude (Mbol) of a star is known, then its luminosity easily can be found as a function of the Sun's luminosity with the relation:

Log (L'/L) = 0.4 (Mbol - Mbol *)

where Mbol is for the Sun and Mbol * is for the star. Note that any given absolute visual magnitude (MV) can be changed to an absolute bolometric magnitude by applying a bolometric correction such that:

Mbol * = MV* + B.C.

Example Problem:

The star Almach (Gamma Andromeda) has (B - V) = +1.3 and an apparent visual magnitude mV* = 2.16. What bolometric correction should be applied? Also, find the absolute visual magnitude MV* and the absolute bolometric magnitude Mbol * of the star. How does it compare in luminosity to the Sun? (The distance of Almach is 80 pc.)

Solution:

From the Table provided:

We find (B - V) = +1.3 corresponds to B.C. = - 0.92.

The absolute visual magnitude can be found from the apparent visual magnitude. Thus:

(m - M) = (m V* - MV*) = 5 log (D) - 5

and:

MV *= mV* - 5 log (D) + 5 = 2.16 - 5 log (8o) + 5

MV* = 2.16 - 5(1.903) + 5 = -2.36

The absolute bolometric magnitude is:

Mbol * = MV* + B.C. = (-2.36) + (-0.92) = -3.28

The relative luminosity as a function of absolute bolometric magnitude is:

Log (L'/L) = 0.4 (Mbol - Mbol *) = 0.4 (4.63 - (-3.28))

Log (L'/L) = 0.4(7.91) = 3.16

antilog 3.16 =1445 so: L' = 1445 L

Suggested Additional Problems:

(1) The apparent V-band (filter) magnitude of a star is 8.72, and it requires a bolometric correction of -0.48. Find the apparent bolometric magnitude of the star. (Hint: Apparent bolometric magnitudes are obtained in an analogous way to the absolute forms)

(2) A star has a color index (B - V) of +1.0 and its apparent B-band magnitude is 6.4. The corresponding bolometric correction is - 0.5. Find the apparent bolometric magnitude of the star.

(3) The star Alhena in the constellation Gemini is at a distance of 30 pc. If it has (B - V) = 0.00, and mV = +1.93, find the apparent B-band magnitude, mB.

Also find the absolute visual magnitude and the absolute bolometric magnitude of the star.

Find the luminosity of Alhena in terms of the solar value.