2.3 Measuring Light
13
2.3 Measuring Light
You will often come across the terms luminosity and Brightness. The luminosity of
an object is the total amount of energy that is emitted across a specific wavelength
range. The brightness is a related but subtly different concept: the brightness of an
object is the amount of energy it emits through a specific wavelength range and over
a solid angle. Brightness is also dependent on the distance from the emitting object
and thus will decline in proportion to the square of said distance.
The absolute magnitude of an object is the brightness it would have at a fixed
reference distance. The point of this is that it allows a quick and valid comparison
with the absolute magnitudes of other objects. For solar system objects, the reference
distance is 1 AU. objects outside the solar system, the absolute magnitude is the
magnitude that the object would have if it were located at a distance of 10 pc.
In general, you should use the term “apparent brightness” to avoid confusion.
You may also come across the term “bolometric luminosity” with respect to brightness. It refers to luminosity or brightness across the entire spectrum.
Stellar spectra broadly approximate those of black bodies. That is, they can be
modelled as perfect absorbers and emitters, and they reflect next to no light. (A littleknown fact about the Sun is that in terms of its reflectivity, it is actually about as
black as coal.) This means that their output follows the pattern given by Planck’s law
B λ (T ) =
2hc
2
λ 5
1
e
hc
λkT − 1
,
(2.3)
where B λ (T ) is the total radiation at wavelength λ and temperature T , h is Planck’s
constant, k is Boltzmann’s constant, and c, as usual, is the speed of light in vacuum.
In general, astronomical bodies emit their light isotropically. This means that they
emit equally in all directions. There is a small handful of exceptions, such as pulsars
and quasars (pulsars emit their radiation in beams). However, the majority of objects
that you encounter will be isotropic radiators. How do we analyse isotropic emitters?
We can begin by considering a sphere of radius r that surrounds an isotropic emitter.
The amount of energy passing through any given area of the sphere will decline
as the sphere gets larger; keep in mind that energy is conserved, so as the sphere
grows, the same amount of energy will be spread over a larger and larger area. This
means that the flux is directly proportional to the emitter’s luminosity and inversely
proportional to the square of the distance between emitter and observer. Hence flux
can be defined as follows:
F =
L
4πr 2 .
(2.4)
This implies that the total amount of flux that a telescope’s aperture can gather is
dependent on the diameter of the telescope. A smaller one will gather less, a bigger
one will gather more. If it helps, think of a telescope as acting for photons as a bucket
does for rainfall.
13
2.3 Measuring Light
You will often come across the terms luminosity and Brightness. The luminosity of
an object is the total amount of energy that is emitted across a specific wavelength
range. The brightness is a related but subtly different concept: the brightness of an
object is the amount of energy it emits through a specific wavelength range and over
a solid angle. Brightness is also dependent on the distance from the emitting object
and thus will decline in proportion to the square of said distance.
The absolute magnitude of an object is the brightness it would have at a fixed
reference distance. The point of this is that it allows a quick and valid comparison
with the absolute magnitudes of other objects. For solar system objects, the reference
distance is 1 AU. objects outside the solar system, the absolute magnitude is the
magnitude that the object would have if it were located at a distance of 10 pc.
In general, you should use the term “apparent brightness” to avoid confusion.
You may also come across the term “bolometric luminosity” with respect to brightness. It refers to luminosity or brightness across the entire spectrum.
Stellar spectra broadly approximate those of black bodies. That is, they can be
modelled as perfect absorbers and emitters, and they reflect next to no light. (A littleknown fact about the Sun is that in terms of its reflectivity, it is actually about as
black as coal.) This means that their output follows the pattern given by Planck’s law
B λ (T ) =
2hc
2
λ 5
1
e
hc
λkT − 1
,
(2.3)
where B λ (T ) is the total radiation at wavelength λ and temperature T , h is Planck’s
constant, k is Boltzmann’s constant, and c, as usual, is the speed of light in vacuum.
In general, astronomical bodies emit their light isotropically. This means that they
emit equally in all directions. There is a small handful of exceptions, such as pulsars
and quasars (pulsars emit their radiation in beams). However, the majority of objects
that you encounter will be isotropic radiators. How do we analyse isotropic emitters?
We can begin by considering a sphere of radius r that surrounds an isotropic emitter.
The amount of energy passing through any given area of the sphere will decline
as the sphere gets larger; keep in mind that energy is conserved, so as the sphere
grows, the same amount of energy will be spread over a larger and larger area. This
means that the flux is directly proportional to the emitter’s luminosity and inversely
proportional to the square of the distance between emitter and observer. Hence flux
can be defined as follows:
F =
L
4πr 2 .
(2.4)
This implies that the total amount of flux that a telescope’s aperture can gather is
dependent on the diameter of the telescope. A smaller one will gather less, a bigger
one will gather more. If it helps, think of a telescope as acting for photons as a bucket
does for rainfall.