The magnitude of a star is a measure of its brightness as seen from Earth. It is logarithmically based on the amount of light that reaches our eyes from a star, with lower magnitudes corresponding to brighter stars and higher magnitudes corresponding to dimmer stars. The magnitude scale was originally developed by the ancient Greek astronomer Hipparchus and has been revised over time.

Apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth normalized to the value it could have in the absence of the atmosphere. Ptolemy (BC 340) introduced a system assigning to star 'a number' depending on its apparent brightness. He suggested '' 1'' for the brightest star and ''6'' for the fainted. This system was based on how bright a star appear to the unadded eye. It is because of the traditional convention, the brighter the object appears, lower the value of its magnitude.

Norman Robert Pogson (1856) studied the perceiveness of human eye to the brightness using a series of candle experiments and concluded that human eye has a logarithmic response towards the brightness

i.e m ∝ log b

If m1 and m2 represent apparent magnitudes of two stars having brightness b1 and b2, then according to Pogson,

m1- m2 ∝ log b1 - log b2

m1 - m2 = k log(b1/b2) .....i

Pogson experimentally found that a difference of 5 magnitudes corresponds to the factor of 100 times in brightness. It means a difference of 5 magnitudes corresponds to a brightness ratio of exactly 100:1. For example, a star with a magnitude of 2 is 100 times brighter than a star with a magnitude of 7. A difference of 1 magnitude corresponds to a brightness ratio of approximately 2.512:1.

The brightest star, Sirius, has a magnitude of -1.46, while the faintest stars that can be seen with the naked eye have a magnitude of about 6.

It is important to note that the magnitude scale is not linear, but logarithmic. This means that a difference of 1 magnitude corresponds to a much greater difference in brightness for dim stars than it does for bright stars.

Numerically,

m1- m2 = 5

then, b1/b2 = 1/100

=> b2 = 100 b1

Substituting in equation i, we get

5 = k log (1/100) = k log (10)^-2 = - 2k

So k = -2.5

Thus equation 1 takes this form,

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

It was assumed that the stars having m1 and m2 are at the same distance. If m1 -m2 = 1 then equation ii gives

log (b1/b2) = - 0.40

Thus, first magnitude star is 2.512 times brighter than second magnitude star. Hence, there is no linear relationship between the magnitude and brightness of stars.

The magnitude system has several advantages for astronomers:

It is a standardized system: The magnitude system allows astronomers to compare the brightness of stars across the entire night sky using a common scale, making it easier to study the stars and their characteristics.
It provides a way to measure distant stars: By measuring the apparent magnitude of a star, astronomers can calculate its absolute magnitude, which is a measure of its intrinsic brightness. This information can be used to determine the distance to the star, which is critical in understanding the structure and evolution of the universe.
It helps classify stars: The magnitude system is used in conjunction with other data, such as a star's temperature and spectral type, to classify stars and understand their physical properties.
It enables us to study variable stars: By observing the changes in the magnitude of a star over time, astronomers can study variable stars and learn about their physical processes.
It has a long history: The magnitude system has a rich history, dating back to ancient Greece, and has been refined and developed over centuries by astronomers. This long history provides a valuable perspective on the study of stars and the universe.

Overall, the magnitude system provides a simple and effective way for astronomers to study stars and their properties, and its use continues to be a fundamental part of modern astronomy.

Theory of random walk

Random walk can be describe statistically. In the star, an individual photon at the core may be absorbed or scattered and then re emitted, absorbed again in many succession. The direction in which photon is emitted bear no relationship at all to the direction in which it was travelling just before absorption. The term "random walk" is used because the direction and magnitude of these perturbations are uncertain and can change over time, causing the star's trajectory to resemble a random walk.

The random walk of stars can also play a role in shaping the structure and evolution of star clusters, galaxies, and the Milky Way itself. In some cases, the random walk of stars can be modelled using statistical methods and simulations, allowing astronomers to study the dynamics of star systems and the effects of different physical processes on star motion. However, due to the many complexities involved and the inherent uncertainty of the underlying processes, the exact trajectory of a star is often difficult to predict with certainty.

Let the step length of the photon is 'd'. Let us consider for simplicity, random walk is in a plane. After one step, the photon is absorbed at

After N steps, the coordinates are,

In general, the magnitude of the displacement in a random walk is related to the number of steps taken and the size of each step. For example, in a simple random walk in one dimension, the mean square displacement is proportional to the square of the number of steps. The distance from the starting point is simply

r² = x² + y²

The second term of equation iii,

The second term is important only when there is relation between consecutive angles.

In the scattering, the second never vanish whereas it vanishes if the process is absorption /emission. Thus we can write,

r² = d² N x 1

r = d √ N .... v

Thus, after N steps, the photon is at the distance d √ N from the starting point. Thus the statistical average of overall displacement r after N steps is simply d √ N .

In general, there is finite possibilities of several types of scattering (Compton, Rayleigh etc), thus the overall displacement is given as

r > d √ N

It refers to a relation between the magnitude of the displacement of the star (r), the step size of the random walk (d), and the number of steps taken (N). In general, this equation provides a rough estimate of the magnitude of the displacement of a star in a random walk, and it can be used as a starting point for more detailed models and simulations. However, the exact behaviour of a star in a random walk can be complex and depends on many factors, including the underlying physical processes and the distribution of mass and matter in the Milky Way and beyond.

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