29 Physical Characteristics of Stars

V. B. Bhatia

1. Learning Outcomes

After studying this module, you should be able to

understand the meaning of apparent magnitude of a star
appreciate the fact that lower magnitude means higher brightness
derive the mathematical relationship between apparent magnitude and brightness
explain the concept of absolute magnitude
relate mathematically absolute magnitude with apparent magnitude and the distance of the star
understand the meaning of distance modulus
derive the mathematical relation between absolute magnitude and luminosity
appreciate distribution of stars in apparent and absolute magnitudes

2. Introduction

In the last two modules, Modules 06 and 07, we started discussing the physical properties of stars. Since determination distances of stars is crucial to getting to the properties like brightness and effective surface temperature, we focused our attention on distance determination. The first step in this direction is the method of parallax, the tiny shift in the direction of the star when observed from two different points. To make this shift measurable, we make observations at time intervals of six months from the opposite ends of the diameter of earth’s orbit round the Sun. It was pointed out that all indirect methods of distance determination depend crucially on the distances determined for the stars whose parallaxes can be measured.

We also introduced the concepts of proper motion, space velocity and peculiar velocity of stars. Since the velocity vectors of all stars in our neighbourhood converge in a certain direction, it was inferred that it is the reflection of Sun’s own motion in the opposite direction. When the velocity of a star is found relative to the motion of the Sun, it is called its peculiar velocity. Peculiar velocities are essentially random so that in a time of the order of 105 years, all constellations will be thoroughly shuffled.

In this module we discuss the brightness of stars, one of their important physical property. We shall introduce concepts like apparent and absolute magnitudes, the latter leading to the knowledge of stellar luminosity, again an important attribute of a star. We shall see that the difference between the apparent and absolute magnitude of a star depends on its distance and thereby it provides an indirect method of distance measurement of the star.

3. Stellar Magnitudes

Hipparchus (second century BCE) was probably the first astronomer who compiled a catalogue of stars. Obviously the catalogue contained only the stars visible to the naked eye. Based on this work, Ptolemy (second century CE) introduced the system of assigning to each star a number depending on its brightness. The numbers ran from 1 to 6, the number 1 being assigned to the brightest star and number 6 being assigned to the faintest star visible: the smaller the number the brighter the star. The same system survives today and is applied not only to stars visible to the naked eye but also to those which are visible through telescopes. The numbers assigned to stars and other astronomical objects to indicate their brightness are referred to as apparent magnitudes, denoted by ?(lower case). With the advent of telescopes and sensitive detectors, the magnitudes are no longer confined to whole numbers; they are now given accurately to second or third place of decimal. Moreover, for some objects, which are too bright to be assigned magnitude 1, zero and negative magnitudes have been introduced. The brightest star in the sky, Sirius, for example, has a magnitude of -1.44.

When it became possible to measure the magnitudes of celestial objects more accurately, it was found that the difference in the magnitudes of two stars is proportional to the logarithm of the ratio of their brightness. This was quite consistent with Fechner’s law (also called Weber – Fechner law) in biology, which stipulates a logarithmic relationship between the apparent magnitude of a sensation (perceived intensity) and the stimulus which produces this sensation.

3.1. Brightness and Apparent Magnitude

10. Summary

Magnitude is a number assigned to a star depending on its brightness.
Higher the brightness of the star, smaller is its magnitude.
Difference of magnitudes of two stars is proportional to the logarithm of their brightness ratio.
These magnitudes are apparent magnitudes.
A difference of one in magnitudes corresponds to a ratio of 2.512 in brightness.
A difference of 5 in magnitude corresponds to a ratio of 100 in brightness. This is Pogson’s law.
With naked eye we can see only stars brighter than 6 magnitude. This number is around 5000.
The number of bright stars in the sky is very small. Towards higher magnitudes the number of stars shoots up.
Apparent magnitudes do not give any clue to the luminosities of stars.
Absolute magnitude of star is defined as its magnitude when placed at a distance of 10 pc.
Difference of apparent and absolute magnitude of a star is called distance modulus, as it gives the distance of the star.
We must compensate for the absorption in the interstellar space, otherwise we tend to overestimate distances.
The absolute magnitude of the Sun is ~ 5.
The most luminous star has luminosity ~ 10⁶ ?⨀, while the least luminous star has luminosity ~ 10^-5 ?⨀.
Difference of absolute magnitudes of two stars is proportional to the ratio of their luminosities.
The number of stars more luminous than the Sun is small. Most stars have luminosities smaller than that of the Sun.

Know More

Net sources for the material covered in this module:

https://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law
http://www.geneseo.edu/~mounts/Courses/Psyc307/Outlines/Ch01%20Lecture%20Part%20II.pdf
http://docs.exdat.com/docs/index-153835.html
http://www.scratchapixel.com/old/lessons/3d-basic-lessons/lesson-14-interaction-light-matter/light-as-a-particle/
http://csep10.phys.utk.edu/astr162/lect/stars/cindex.html
http://www.skyandtelescope.com/astronomy-resources/the-stellar-magnitude-system/