Introduction:

A fundamental observable quantity for a star is its brightness. As stars can encompass a very wide range of brightness, astronomers generally introduce a logarithmic scale termed as a magnitude scale to categorize the brightness.

The magnitude scale:

The magnitudes m₁ and m₂ for two stars are associated to the corresponding brightness b₁ of star m₁and b₂ of star m₂ via the equation:

m₂ - m₁ = 2.5 (log b₁ - log b₂) = 2.5 (b₁/b₂)

Here log signifies the (base-10) logarithm of the corresponding number; that is, the power to which 10 should be raised to give the number. As this relation is logarithmic, a very big range in stellar brightness corresponds to a much smaller range of magnitudes; this is the main utility of the magnitude scale. 

If a star of magnitude 1 is 2.512 × brighter than the star of magnitude 2 and 100 × brighter than the sixth magnitude star, then how much brighter is it than a star of magnitude 3?

You must be careful here. This is not simply 2 × 2.512 different. You must to keep in mind that a difference of one magnitude equivalents 5√100 = 2.512. A difference of 2 magnitudes thus = 2.5122 = 6.31 × difference in brightness. 

The two objects of various magnitudes thus differ in brightness through 2.512 raised to the power of their magnitude difference. If we write this as an equation, then the ratio of brightness or intensity, b₁/b₂ between two objects, A and B, having magnitudes m₁ and m₂ is represented by the given equation:

b₁/b₂ = (2.512) ^m2-m1

Here b₁ and m₁ are the brightness and magnitude of the star 1 and where b₂ and m₂ are the brightness and magnitude of the star 2.

For illustration, the apparent magnitude of the Sun is around -26. The brightest star in the sky, Sirius, consists of an apparent magnitude of -1.4. The difference in magnitude is around 25; for each and every five magnitudes the brightness ratio is 100. Therefore Sirius is 10² x 10² x 10² x 10² x 10² = 1010 times less bright than the sun in the apparent magnitude. The result states us nothing regarding the luminosities of the Sun and Sirius, only how bright they appear in the sky. In contrary, the absolute magnitude of the sun is around + 4.8, and Sirius is +1.4. The difference in absolute magnitudes is around 4, corresponding to the brightness ratio of around 40. This comparison of absolute magnitudes states us that the Sirius is around 40 times as luminous as the Sun.

By using the above information, we can work out ratio of the brightness of Sirius-Sun as:

b1/b2 = (2.512)^m2-m1

b1/b2 = (2.512)^4.8-1.4

b1/b2 = (2.512)^3.4

b1/b2 = 2

An identical comparison can be made to determine the luminosities of other stars as brightness and intensity are proportional to one other.

Apparent Magnitude:

The prior equation provides us a manner to relate the brightness and magnitudes of the two objects, however there are some ways in which we could identify the brightness and this leads to some different magnitudes which astronomers define. One significant distinction is between whether we are talking about the apparent brightness of an object, or its true brightness. The earlier is a convolution of the true brightness and the effect of distance on the observed brightness, as the intensity of light from the source reduces as the square of the distance (that is, the inverse square law). 

The apparent magnitude of the object is: 'What you see is what you get' magnitude. This is determined by employing the apparent brightness as noticed, without consideration given to how distance is affecting the observation. Of course, the apparent magnitude is simple to find out as we only require measuring the apparent brightness and transform it to a magnitude having no further thought given to the matter. Though, the apparent magnitude is not as helpful as it mixes up the intrinsic brightness of the star (that is associated to its internal energy production) and the effect of distance (that has nothing to do with the intrinsic structure of the star).

Absolute Magnitude:

Evidently, a star which is very bright in our sky could be bright mainly as it is much close to us (the Sun, for illustration), or because it is instead far-away however is intrinsically very bright (Betelgeuse, for illustration). This is the 'true' brightness, having the distance dependence factored out that is of most interest to us as astronomers. Thus, it is helpful to set up a convention whereby we can compare the two stars on the similar footing, without variations in brightness due to differing distances making difficult the issue.

Astronomers state the absolute magnitude to be the apparent magnitude that a star would contain if it were (in our thoughts) placed at a distance of 10 parsecs (that is 32.6 light years) from the Earth.

We can do this if we are familiar with the true distance to the star as we can utilize the inverse square law to find out how its apparent brightness would change if we moved it from its true position to a standard distance of 10 parsecs. There is nothing magical regarding the standard distance of 10 parsecs. We could also employ any other distance as a standard, however 10 parsecs is the distance astronomers have selected for this standard. A general convention and one which we will mostly follow, is to employ a lower-case 'm' to represent an apparent magnitude and an upper-case 'M' to represent an absolute magnitude.  

The absolute magnitude 'M' of a star is stated as the magnitude that star would have if it were at a distance of 10 parsecs from us.

A distance of 10 parsec is purely random however now internationally agreed upon by astronomers. The scale for absolute magnitude is similar as that for apparent magnitude, that is a difference of 1 magnitude = 2.512 times difference in the brightness. This logarithmic scale is as well open-ended and unit less. Again, the lower or extra negative the value of M, the brighter the star is. The absolute magnitude is a convenient means of representing the luminosity of a star. Once the absolute magnitude of a star is recognized you can as well compare it to other stars. Betelgeuse, M = -5.6 is intrinsically more luminous than Sirius by an M = 1.41. Our Sun consists of an absolute visual magnitude of 4.8.

Distance Modulus:

As we know that most of the stars are too distant to have their parallax measured directly. However if we recognize both the apparent and absolute magnitudes for a star we can find out its distance. Let us look at Sirius and Betelgeuse plus another star termed as GJ 75.

How far away is GJ 75? This is an unusual star in that it's apparent and absolute magnitudes are similar. Why? The reason is that it is in reality 10 parsecs distant from us, thus by definition its two magnitudes should be similar.

What about Sirius? Its apparent magnitude is lower (thus brighter) than its absolute magnitude. This signifies that it is closer than 10 parsecs to us. Betelgeuse's apparent magnitude is higher (thus dimmer) than its absolute magnitude therefore it would appear even brighter in the night sky if it were merely 10 parsecs distant.

Is there a fast means of checking whether a star is close or not? Observe the above table we see that a star is at a distance of 10 parsecs, then m = M or m - M = 0. 

For Sirius, m - M = (-1.44) - 1.41 = - 2.85. This value is negative and Sirius is closer than 10 parsec. For Betelgeuse, m - M = 0.45 - (- 5.14) = 5.59. This value is positive and Betelgeuse is more than 10 parsec distance. Astronomers make use of the difference between the apparent and absolute magnitude, the distance modulus, as a manner of finding out the distance to a star.

• Distance Modulus = m - M.
• Distance modulus is negative for the stars nearby than 10 parsecs.
• Distance modulus is positive for the stars further away than 10 parsecs.
• The size of the distance modulus finds out the actual value of the distance, in such a way that a star of distance modulus 1.5 is closer than one by a distance modulus of 8.7.

The distance modulus can be employed to find out the distance to a star by employing the equation:

m - M = 5 log (d/10)

Here, 'd' is in parsecs. Note that if d = 10 pc then m and M are similar. If we make 'M' the subject matter, we have: 

M = m - 5 log (d/10)

It must be noted that we can find out the apparent magnitude 'm' of a star simply by computing how bright it appears to be, however to find out the absolute magnitude 'M' the distance to the star should as well be known.

Influence of Wavelength on Brightness:

You may think that introducing the apparent and absolute magnitudes would resolve the ambiguities regarding what we mean if we refer to the brightness of a star, however there is a further complication. The brightness of an object (that is, whether apparent or absolute) based on the wavelength at which we observe it.

Usually, astronomical observations are made by an instrument which is sensitive to a specific range of wavelengths. For illustration, if we observe by the naked eye, we are sensitive only to the visible portion of the spectrum, by the most sensitivity coming in the yellow-green part of that. On the other hand, if we employ normal photographic film to record our observation, this is more sensitive to blue light than to yellow-green light.

Brightness - Luminosity Relationship:

This associates the Apparent Brightness of a star (or other light source) to its Luminosity (that is, Intrinsic Brightness) via the Inverse Square Law of Brightness and it is represented by :

B = L/4πd²

Here, 'B' is the brightness of the star; 'L' is the Luminosity and 'd' the distance from the source.

At a specific Luminosity, the more distant an object is, the fainter its apparent brightness becomes as the square of the distance. To compute the Luminosity of a star we require taking two measurements in account viz:

• The Apparent Brightness (that is, flux) measured through photometry.
• The Distance to the star measured in some manner.

Therefore altogether by the inverse square law of brightness, we can calculate the Luminosity.

Measuring Apparent Brightness:

The method of measuring the apparent brightness of objects is termed as Photometry. The two modes to deduce apparent brightness are:

1) Stellar Magnitudes

2) Absolute Fluxes (energy per second per area)

Flux Photometry:

This is a tool that is employed to compute the photons received from a star by employing a light-sensitive detector. A few of the detectors are:

• Photographic Plates
• Photoelectric Photometer
• Solid State Detector (example: photodiodes and CCDs)

We now employ solid-state detectors such as CCDs and identical technologies (having very rare exceptions), as such detectors are far more sensitive and stable than any prior technology.

Luminosity of Stars:

The absolute magnitude of a star is simply a simple means of explaining its luminosity. Luminosity 'L' is the measure of the net amount of energy radiated through a star or other celestial object per second. This is thus the power output of a star. A star's power output across all wavelengths is termed its bolometric luminosity.

Our Sun consists of a luminosity of 3.84 × 10²⁶ W or J.s^-1 that can be represented by the symbol L_sol (in reality the subscript symbol is generally a dot within a circle - the standard astrological symbol for the Sun). Instead of always use this precise value it is often more suitable to compare the other star's luminosity L* to the Sun's as a fraction or multiple. Therefore if a star is twice is luminous as the Sun, L*/L_sol = 2. This approach is suitable as the luminosity of stars differs over a vast range from less than 10^-4 to around 10⁶ times that of the Sun so an order of magnitude ratio is often adequate. Luminosity is found out by:

1) Temperature: The black body radiates power at a rate associated to its temperature - the hotter the black body, the greater its power output per unit surface area. The incandescent or filament light bulb is an everyday illustration. As it gets hotter it gets brighter and releases more energy from its surface. The association between power and temperature is not a simple linear one though. The power radiated via a black body per unit surface area differs by the fourth power of the black body's effective temperature, T_eff.

Therefore; the power output, l is directly proportional to T⁴ or l = σT⁴ for a perfect black, here σ is a constant termed as the Stefan-Boltzmann constant. It consists of the value of σ = 5.67 × 10^-8 W m^-2 K^-4 in SI units. As star is not a perfect black body we can estimate this relationship as:

I ≈ σT⁴

This relationship helps account for the vast range of stellar luminosities. A small raise in efficient temperature can significantly raise the energy emitted per second from every square metre of the surface of star.

2) Size (radius): If two stars encompass the similar effective temperature however one is bigger than the other it consists of more surface area. The power output per unit surface area is fixed via in such a way that the star having greater surface area should be intrinsically more luminous than the smaller one.

Supposing stars are spherical then the surface area is represented by:

Surface area = 4πR²

Here, 'R' is the radius of the star

To compute the net luminosity of a star we can join the equations above to give:

L ≈ 4πR²σT⁴

Comparing Luminosities and Brightness:

Let us suppose we have two stars, A and B which we wish for to compare. If we can measure their individual apparent magnitudes, mA and mB how will they differ in brightness? The ratio of their brightness (or intensities) I_A/I_B corresponds to their difference in magnitude, m_B - m_A. Keep in mind, as a difference of one magnitude signifies a brightness ratio of the fifth root of 100 or (100)^1/5, a difference of m_B - m_A magnitudes provides a ratio of [(100)^1/5]^{mB - mA}

∴ I_A/I_B = 100^(mB-mA)/5

The inverse-square law of light signifies that the flux, 'l' (or intensity) of a star at a distance 'd' can be associated to its luminosity 'L' at a distance 'D' by the given relationship:

L/l = (d/D)² = (d/10)²

At distance of 10 parsecs, 'D' is symbolized by absolute magnitude, 'M' and the flux at distance 'd' is represented through the apparent magnitude, 'm' then the luminosity ratio is represented by:

m - M = 2.5 log (L/l)

m - M = 2.5 log (d/10)²

m - M = 5 log (d/10)

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