Transcript Tutorial: Luminosity

Measurements of Luminosity
For a refresher on Stefan’s Law, upon
which this tutorial is based, please
consult the Tutorial on Continuous
Spectra
Luminosity (brightness) of a Star
Luminosity is the amount of energy per second (Watts) emitted by the star
Recall:
The luminosity of the sun is about 4 x 1026 W
Absolute Brightness: The luminosity per square meter emitted by the
star at it’s surface. This is an intrinsic property of the star.
Apparent Brightness: The power per square meter as measured at the
location of the earth.
Luminosity (brightness) of a Star
How it works: Photons flow radially outward from the center of the star. The
luminosity is the energy output per second, that is, the power output from
the star. The luminosity, therefore, is a constant for a given star.
Direction of photon
(energy) flow
(radially outward)
Luminosity (brightness) of a Star
However, the “brightness” of a star decreases as one moves farther and
farther away.
Reason: The energy/sec flows from the interior of the star through its
surface. The surface of the star is a sphere, with a surface area given by
4πR2 , where R is the radius of the star.
Direction of photon
(energy) flow
(radially outward)
R
Luminosity (brightness) of a Star
However, the “brightness” of a star decreases as one moves farther and
farther away.
If a sphere of radius d is drawn around the star, it should be clear that the
energy/sec through the surface of this sphere is the same as the energy/sec
emitted through the surface of the star, since there is no mechanism to
create or destroy photons in the space outside the star. The surface of the
sphere has an area given by 4πd2 , where d is the radius of the sphere.
Direction of photon
(energy) flow
(radially outward)
d
Luminosity (brightness) of a Star
Absolute Brightness: The luminosity per square meter emitted by the
star at it’s surface. This is an intrinsic property of the star.
Apparent Brightness: The power per square meter as measured at a
distance d from the star. If the observation of brightness of the star is
made from the earth, the apparent brightness is the power per square
meter as measured from the earth. d would then be the distance from
the earth to the star.
Direction of photon
(energy) flow
(radially outward)
L (the Luminosity)
is the SAME in
both expressions.
R
Ab. Bright = L / 4 π R2
Where R is the radius
of the star
d
App. Bright = L / 4 π d2
Where d is the distance
to the observer
Luminosity (brightness) of a Star
Prove that the luminosity of the sun is 4 x 1026 W.
From Appendix 2,
Radius of earth’s orbit = 1.496 x 1011 m
Also, the magnitude of the sun (Pogson scale) is -26.7. Now, there is a
footnote in the chapter (you would not need to know this) that a
magnitude 1 star has an apparent brightness of 1.1 x 10-8 W/m2 . Recall
that each step in magnitude corresponds to a brightness 2.512 times
greater than the lower magnitude.
So, a magnitude 0 is 2.512 times greater than a magnitude 1
A magnitude -1 is (2.512)2 times greater than a magnitude 1
A magnitude -2 is (2.512)3 times greater than a magnitude 1
And continuing the logic,
A magnitude -27 is (2.512)28 times greater than a magnitude 1
Luminosity (brightness) of a Star
Prove that the luminosity of the sun is 4 x 1026 W.
From Appendix 2,
Radius of earth’s orbit = 1.496 x 1011 m
A magnitude -26.7 is a bit less than (2.512)28 times greater than a
magnitude 1. To make the calculation simpler, we will use -27 for the
magnitude of the sun.
Apparent Brightness of the sun
= (2.512)28 (1.1 x 10-8 W/m2 )
= (1.59 x 1011 ) (1.1 x 10-8 W/m2 )
= 1749 W/m2
Luminosity (brightness) of a Star
Prove that the luminosity of the sun is 4 x 1026 W.
From Appendix 2,
Radius of earth’s orbit = 1.496 x 1011 m
And finally
Apparent Brightness = 1749 W/m2 = L / 4πr2
So,
L
= ( 1749 W/m2 ) 4 π (1.496 x 1011 m)2
= 4.9 x 1026 W
Now, since the magnitude of the sun is actually -26.7, using a magnitude
of 27 is an overestimate, and the calculated luminosity is a bit too high,
as expected.
Luminosity (brightness) of a Star
Two stars have the same apparent brightness as measured from the earth.
Star A is 15 pc from the earth, star B is 30 pc from the earth. Which star
has the greater luminosity, and how much greater is the luminosity of this
star as compared to the lower luminosity star?
Apparent Brightness = L / 4πd2
Since the two stars have the same apparent brightness, the star that
is further from the earth (star B) must have the higher luminosity.
Since
(App Bright)A = (App. Bright)B
LA / 4πd A 2 = LB / 4πdB 2
LB = LA (dB / dA )2 = LA (30 pc / 15 pc)2 = LA (2)2 = 4 LA
The Luminosity of Star B is 4 times greater than the
Luminosity of Star A.