# Star Light, Star Bright

By

Dr Lim Zhi Han

,

Koh Zhi Yong

,

Gerald Kang Joon Kiat

●

Published 2022-09-10

You have probably noticed that far-away lights seem dimmer than those closer to you. This is because the light intensity decreases as the square of the distance between you and the light. The energy emitted by a light source per unit time, or power, is determined by the light source itself. Assuming the light is emitted isotropically ¹, as the light waves travel away from the source, they are spread out over larger surface areas. As such, the light energy passing through a location per unit time per unit area, decreases with 1/d^{2}. Since your eye, the light detector, is the same size regardless of how far you are from the light source, the light-capturing area is the same. But due to the reduced irradiance further away, the total energy per unit time caught by your eye is reduced, leading to the perception of a dimmer light.

A similar phenomenon is at play when we observe stars in the night sky. Stars are basically spherical light sources far away from us that emit light isotropically.

Go to Activity 1.

The intrinsic luminosity L of a star quantifies how much light comes out of a source. It is measured in energy emitted per unit time.

The radiation flux F quantifies how much light is received by a detector. It measures the light energy per unit time per unit area.

The two quantities above are related by

F=\frac{L}{4\pi d^{2}}

where d is the distance from the source to the detector.

Strictly speaking, the flux from a star at any point can be defined to be dependent on the distance between the star and that point. However, for the purposes of observational astronomy, we are content to constrain our definition such that the point of observation is Earth.

In observational astronomy, we measure the flux from the star as it reaches our detectors. The flux is quantified in a logarithmic scale called magnitude. This scale eases the process of comparison between different stars. The relative brightness between two stars, as observed on Earth, is quantified by apparent magnitude m, defined as follows

\frac{F_{2}}{F_{1}}=100^{(m_{1}-m_{2})/5}

where F_{1}, F_{2}, are the radiant flux of star 1 and star 2 respectively, and m_{1}, m_{2} are the apparent magnitude of star 1 and star 2 respectively. Note that by this definition, a brighter star will have a lower apparent magnitude, and that it is possible to have a negative magnitude. Also note that this definition merely establishes a way to compare relative brightness, but a zero point is not explicitly set by this equation. By convention, a star called Vega is taken as the zero point of the apparent magnitude scale. From there, the absolute magnitude of Vega is calculated (0.57) which indirectly defines the zero point of the absolute magnitude scale.

The apparent magnitude is a measure of the flux from a star as measured from Earth. Consequently, its measured value depends on the star's distance away from us. This additional confounding factor can be troublesome when we wish to discuss the properties of the star itself, and not how the star appears to us. Thus, we further define the absolute magnitude M as what the apparent magnitude would be, if the star were exactly 10 pc away from us.

\begin{align}
100^{(m-M)/5} & =\left(\frac{d}{10\text{ pc}}\right)^{2}\nonumber \\
10^{(m-M)/5} & =\frac{d}{10\text{ pc}}\nonumber \\
\frac{m-M}{5} & =\log_{10}\left(\frac{d}{10\text{ pc}}\right)\nonumber \\
m & =M+5\log_{10}\left(\frac{d}{10\text{ pc}}\right)
\end{align}

where d is the distance between the star and us, measured in parsecs (pc).

Table 5.1: Table of magnitude of brightness of notable celestial objects. Fill in the Absolute Magnitude column and check your values.

So far, we have discussed the luminous properties of stars in general. Practically, we must also consider that light can have many different wavelengths. No star emits monochromatic light, nor do any emit light of uniform intensity across all wavelengths of the electromagnetic spectrum. Our detectors, however often operate in narrow bands of wavelengths. To account for the fact that stars emit light across a wide range of wavelengths, we define the apparent bolometric magnitude and absolute bolometric magnitude by applying a bolometric correction to the measured apparent magnitude and absolute magnitude respectively.

\begin{align*}
m_{bol} & =m_{V}+BC\\
M_{bol} & =M_{V}+BC
\end{align*}

where the subscript V stands for "visual" and bol stands for "bolometric".

The correction serves to account for light emitted outside of visible wavelengths. The more a star emits in, for example, ultraviolet or infrared wavelengths, the more negative the correction is (i.e. the bolometric magnitude is less than the apparent magnitude, indicating the star is "brighter" than it appears to us).