BRIGHTNESS, RADIANT FLUX AND LUMINOSITY IN ASTRONOMY
BRIGHTNESS, RADIANT FLUX AND LUMINOSITY
Absolute magnitude of astronomical objects
Star cluster showing distance modulus as a measure of distance. For the star farther away, m = 12.3, M = 2.6, r = 871 pc. For the closer star, m = 8.0, M = 5.8, r = 28 pc
It is a common experience that if we view a street lamp from nearby, it may seem quite bright. But if we see it from afar, it appears faint. Similarly, a star might look
bright because it is closer to us. And a really brighter star might appear faint because it is too far. We can estimate the apparent brightness of astronomical objects easily, but, if we want to measure their real or intrinsic brightness, we must take their distance into account. The apparent brightness of a star is defined in terms of what is called the apparent magnitude of a star.
APPARENT MAGNITUDE
In the second century B.C., the Greek astronomer Hipparchus was the first astronomer to catalogue stars visible to the naked eye. He divided stars into six classes, or apparent magnitudes, by their relative brightness as seen from Earth. He numbered the apparent magnitude (m) of a star on a scale of 1 (the brightest) to 6 (the least bright). This is the scale on which the apparent brightness of stars, planets and other objects is expressed as they appear from the Earth. The brightest stars are assigned the first magnitude (m = 1) and the faintest stars visible to the naked eye are assigned the sixth magnitude (m = 6).
Apparent Magnitude
Apparent magnitude of an astronomical object is a measure of how bright it appears. According to the magnitude scale, a smaller magnitude means a brighter star.
The magnitude scale is actually a non-linear scale. What this means is that a star, two magnitudes fainter than another, is not twice as faint. Actually it is about 6.3 times fainter. Let us explain this further.
The response of the eye to increasing brightness is nearly logarithmic. We, therefore, need to define a logarithmic scale for magnitudes in which a difference of 5 magnitudes is equal to a factor of 100 in brightness. On this scale, the brightness ratio corresponding to 1 magnitude difference is 100^1/5 or 2.512.
Therefore, a star of magnitude 1 is 2.512 times brighter than a star of magnitude 2.
It is (2.512)^2 = 6.3 times brighter than a star of magnitude 3.
How bright is it compared to stars of magnitude 4 and 5?
It is (2.512)^3 = 16 times brighter than a star of magnitude 4.
And (2.512)^4 = 40 times brighter than a star of magnitude 5.
expected, it is 2.512^5 = 100 times brighter than a star of magnitude 6.
For example, the pole star (Polaris, Dhruva) has an apparent magnitude +2.3 and the star Altair has apparent magnitude 0.8. Altair is about 4 times brighter than Polaris. Mathematically, the brightness b1 and b2 of two stars with corresponding magnitudes m1 and m2 are given by the following relations.
Relationship between brightness and apparent
Brightness ratio corresponding to given magnitude difference
Modern astronomers use a similar scale for apparent magnitude. With the help of telescopes, a larger number of stars could be seen in the sky. Many stars fainter than the 6th magnitude were also observed. Moreover, stars brighter than the first magnitude have also been observed.
Thus a magnitude of zero or even negative magnitudes have been assigned to extend the scale. A star of −1 magnitude is 2.512 times brighter than the star of zero
magnitude. The brightest star in the sky other than the Sun, Sirius A, has an apparent magnitude of − 1.47.
The larger magnitude on negative scale indicates higher brightness while the larger positive magnitudes indicate the faintness of an object.
The faintest object detectable with a large modern telescope in the sky currently is of magnitude m = 29.
Therefore, the Sun having the apparent magnitude m = − 26.81, is 10^22 times brighter than the faintest object detectable in the sky.
Apparent magnitudes of some celestial objects in the night sky
The apparent magnitude and brightness of a star do not give us any idea of the total energy emitted per second by the star. This is obtained from radiant flux and the luminosity of a star.
Luminosity and Radiant Flux
The luminosity of a body is defined as the total energy radiated by it per unit time.
Radiant flux at a given point is the total amount of energy flowing through per unit time per unit area of a surface oriented normal to the direction of propagation of radiation.
The unit of radiant flux is erg s−1 cm−2 and that of luminosity is erg s−1.
In astronomy, it is common to use the cgs system of units. However, if you wish to convert to SI units, you can use appropriate conversion factors.
Note that here the radiated energy refers to not just visible light, but includes all wavelengths.
The radiant flux of a source depends on two factors:
(i) the radiant energy emitted by it, and
(ii) the distance of the source from the point of observation.
Suppose a star is at a distance r from us. Let us draw an imaginary sphere of radius r round the star. The surface area of this sphere is 4πr^2. Then the radiant flux F of the star, is related to its luminosity L as follow.
The luminosity of a stellar object is a measure of the intrinsic brightness of a star. It is expressed generally in the units of the solar luminosity, LΘ, where
Now, the energy from a source received at any place, determines the brightness of the source. This implies that F is related to the brightness b of the source: the brighter the source, the larger would be the radiant flux at a place. Therefore, the ratio of brightness in Eq...2 can be replaced by the ratio of radiant flux from two objects at the same place and we have
Eq... 5
You know from Eq...3 that the flux received at a place also depends on its distance from the source. Therefore, two stars of the same apparent magnitude may not be equally luminous, as they may be located at different distances from the observer: A star’s apparent brightness does not tell us anything about the luminosity of the star.We need a measure of the true or intrinsic brightness of a star. Now, we could easily compare the true brightness of stars if we could line them all up at the same distance from us. With this idea, we define the absolute magnitude of a star as follows:
Absolute Magnitude
The absolute magnitude, M, of an astronomical object is defined as its apparent magnitude if it were at a distance of 10 pc from us.
Let us now relate the absolute magnitude of a star to its apparent magnitude. Let us consider a star at a distance r pc with apparent magnitude m, intrinsic brightness or luminosity L and radiant flux F1. Now when the same star is placed at a distance of 10 pc from the place of observation, then its magnitude would be M and the corresponding radiant flux would be F2. From Eq...5 we have
Using Eq...6 in Eq...7 we get the difference between the apparent magnitude (m) and absolute magnitude (M).
It is a measure of distance Astronomical Scales and is called the distance modulu
Eq...8
We can also relate the absolute magnitudes of stars to their luminosities. From Eq...3 , we know that the ratio of radiant flux of two stars at the same distance from the point of observation is equal to the ratio of their luminosities. Thus, if M1 and M2 are the absolute magnitudes of two stars, using Eq...6 , we can relate their luminosities to M1 and M2.
Relationship between Luminosity and Absolute Magnitude
Eq...9 and Eq...10
Thus, the absolute magnitude of a star is a measure of its luminosity, or intrinsic brightness.
Often if we know what kind of star it is, we can estimate its absolute magnitude. We can measure its apparent magnitude (m) directly and solve for distance using
Eq...8 For example, the apparent magnitude of Polaris (pole star) is +2.3. Its absolute magnitude is −4.6 and it is 240 pc away. The apparent magnitude of Sirius A is −1.47, its absolute magnitude is +1.4 and it is at a distance of 2.7 pc.
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