Transcript Chapter 3 Continuous Spectrum of Light 3.1 Stellar Parallax d(pc) = 1/p” where p” is the parallax in arc seconds d(pc) is the distance in.

Chapter 3
Continuous Spectrum of Light
3.1 Stellar Parallax
d(pc) = 1/p”
where p” is the parallax in arc seconds
d(pc) is the distance in parsecs
1 pc =3.26 light years and is the distance
at which a star would have a parallax of 1”
Stellar Parallax
Chapter 3.2
The Magnitude Scale
Apparent Magnitude – Invented by Hipparchus. The
brightest stars were 1st magnitude and the faintest, 6th
magnitude. The lower the number, the brighter the
star.
Norman Pogson discovered that a difference of 5
magnitudes corresponds to a factor of 100 in
brightness. A difference of 1 magnitude corresponds
to a brightness ratio of 1001/5 ~ 2.512.
The Magnitude Scale
Apparent Magnitude – A first magnitude star is 2.512
times brighter than a 2nd magnitude star, etc. Apparent
magnitude, m, for the sun -26.83. Faintest object
detectable, m = 30. Faintest star a person with 20/20
vision can see is ~6th magnitude.
The Magnitude Scale
Absolute Magnitude – This is the apparent magnitude
a star would have if it were 10 parsecs or 32.6 light
years distant.
Radiant Flux
A star’s brightness is the total amount of starlight
energy/second at all wavelengths (bolometric) incident
on one square meter of detector pointed at the star.
This is called the radiant flux, F and is measured in
watts/m2.
Luminosity
Luminosity, L is the total amount of energy emitted by
a star into space.
Inverse Square Law
The relationship between a star’s luminosity L, and its
radiant flux, F is:
L
F=
4p r 2
where r is the distance to the star.
Inverse Square Law
Knowing the values for the sun’s luminosity, L and
using 1 AU for the average Earth-sun distance, for r,
the solar flux, F received by the Earth from the sun is:
F = 1365 W/m2,
This is also known as the solar constant.
Distance Modulus
A measure of the distance to a star is called its distance
modulus, or m – M.
m – M = 5log10(d) – 5 = 5 log10 (d/10 pc)
Remember, that M is the magnitude a star would have
if it were 10 pc (parsecs) distant.
Distance Modulus
The distance modulus is a logarithmic measure of the
distance to a star. Here’s a listing of m, M, and m – M
of notable stars:
Star Name
m
M
m–M
Sirius
-1.5
1.4
-2.9
Alpha Centauri A
0.0
4.4
-4.4
Alpha Centauri B
1.4
5.8
-4.4
Proxima Centauri
10.7
15.1
-4.4
Sun
-26.75
4.83
-31.58
Distance Modulus
Example to calculate the M of Proxima Centauri:
m – M = 5 log10 (d/10 pc)
M = m - 5 log10 (d/10 pc)
For Proxima Centauri, d = 1.295 pc, m = 10.7 (it’s a
red dwarf).
M = 10.7 - 5log10 (1.295/10) = 15.1
Chapter 3.4
Blackbody Radiation
A blackbody is defined as an object that does not
reflect or scatter radiation shining upon it, but absorbs
and reemits the radiation completely. It is a kind of
ideal radiator, which cannot really exist. However, stars
and planets behave as if they were blackbodies.
Blackbody radiation depends only on its temperature
and is independent of its shape, material and
construction.
Latest Spectrum of SN2014J
Blackbody Radiation
This figure shows the blackbody spectrum of an object
has a certain peak, and goes to shorter wavelengths as
the temperature increases.
The Planck Function
2hc 2 / l 5
Bl (T ) = hc/lkT
e
-1
where h = Planck’s constant = 6.626 x 10-34J s
and the units of Bl(T) are W m-3 sr-1
Wien’s Law
Notice that the peak of a BB curve goes to shorter and
shorter wavelengths as T increases. This is Wien’s Law.
lmaxT = 0.0029 m K
Chapter 3.4
Blackbody Radiation
Remember this animation site?
http://www.astro.ubc.ca/~scharein/a311/Sim.html
It has a great blackbody simulation/animation!
Wien’s Law
Example: 3.4.1
What is lmax for Betelgeuse? Its temperature is 3600 K.
0.0029mK
lmax @
º 8.05x10-7 m = 805nm
3600K
which is in the IR.
Wien’s Law
Example: 3.4.1
Similarly for Rigel. What is lmax for Rigel? Its
temperature is 13,000 K.
0.0029mK
lmax @
º 2.23x10-7 m = 223nm
13, 000K
which is in the UV.
Chapter 3.6
Color Index
The Johnson - Morgan UBV Filter System
Chapter 3.6
Color Index
The Johnson - Morgan UBV Filter System
Approximate central wavelengths and bandwidths are:
Band
< λ > ( A)
U
3600
560
B
4400
990
V
5500
880
∆λ ( A)
Color Index
Color index is related to a star’s temperature. Figure below is
from
http://csep10.phys.utk.edu/astr162/lect/stars/cindex.html
Returning to The Planck Function
2hc / l
Bl (T ) = hc/lkT
e
-1
2
5
where h = Planck’s constant = 6.626 x 10-34J s
and the units of Bl(T) are W m-3 sr-1
Exploring the Planck Function
Let’s investigate this function.
2hc 2 / l 5
Bl (T ) = hc/lkT
e
-1
The exponent in the denominator is sometimes written as b.
And b = hc/lkT, has no units, ie. it is dimensionless.
h = 6.626068 x 10-34 joule sec
k =1.38066 x 10-23 joule deg-1
c =2.997925 x 108 m/s
T = object temperature in Kelvins
Example Using the Planck Function
Let’s do an example with this function. Consider an object
at 213 K and let’s compute the emitted radiance at 10
microns. Everything is in SI units, i.e. meters and joules.
We have that b =,
(6.63x10-34 Js)(3.0x108 m / s
b=
= 6.77
-23
-5
(1.38x10 J / K)(1x10 m)(213K)
Notice that all of the units cancel.
The denominator of the Planck function is e6.77-1 = 870.3
Example Using the Planck Function
The numerator becomes
2(6.63×10−34 Js)(3.0×108 m/s)2 (1.0×10−5 m)−5
=119.3×107 Jm−3s−1steradian−1
The full Planck function becomes
B =119.3×107 /870.3 = 0.137×107 Jm−3s−1steradian−1
and the units of Bl(T) are W m-3 sr-1
Back to Color Indices and the Bolometric Correction
A star’s U – B color index is the difference between its UV
and blue magnitudes.
A star’s B – V color index is the difference between its blue
and visual magnitudes.
The difference between a star’s bolometric magnitude and
its visual magnitude is its bolometric correction BC.
A bolometric magnitude includes extra flux emitted at other
wavelengths. The magnitude adjustment that takes into
account this extra flux is the bolometric correction.
mbol – V = Mbol – MV = BC
Color Indices and the Bolometric Correction
One can associate an absolute magnitude with each of
the filters.
U – B = M U – MB
B – V = MB – MV
The difference between a star’s bolometric magnitude
and its visual magnitude is:
mbol – V = Mbol – MV = BC
Bolometric Correction
The significance of the bolometric correction is that, the
greater its value, either positive or negative, the more
radiation the star is emitting at wavelengths other than
those in the visible part of the spectrum.
This is the case with very hot stars, where most of their
radiation is in the UV. The BC would be negative.
This is also the case with very cool stars, where most of their
radiation is in the IR. The BC would be posative.
The BC is least with sun-like stars, since most of their
radiation is in the visible part of the spectrum. The BC
would ~ 0.
Example Using Bolometric Correction
For Sirius, U = -1.47, B = -1.43, V = -1.44
U – B = -1.47 –(-1.43) = -0.04
B – V = -1.43 – (-1.44) = 0.01
This shows that Sirius is brightest at UV wavelengths, which is what
you would expect, since T = 9970 K.
0.0029mK
= 291nm
9970K
which is in the UV part of the electromagnetic spectrum. The
bolometric correction for Sirius is BC = -0.09. Apparent bolometric
magnitude is:
lmax =
mbol = V + BC = -1.44 + (-0.09) = -1.53
BC is negative for a hot star like Sirius. BC is positive for a cool star
like Betelgeuse.