fisica

1. 1.6. Colors and Temperatures of Stars
Stellar Radiation:
For a star with a radius R,
•total luminosity L is the total energy radiated per unit time in all directions and in all
wavelengths
•Lλλor Lνν is the corresponding monochromatic quantity
•Stellar flux F(R)
4
2
4
)( TB
R
L
RF σπ
π
===
The total luminosity L 42
4 TRL σπ=
A Stellar radiation is not exactly a
black body
42
4 eTRL σπ= Te is effective temperature

2. Measuring starlight with a telescope having area A
Aθθ
R
The energy striking the lens per second Ω= ∫ dIAE αcos
αα is the angle between the direction of the light and the normal to the lens, 1cos ≈α
2
cos
r
d
d
θ∑
=Ω
r
∑d is the element of surface area of the star, θθ is the angle between the normal to dΣΣ and
the direction to Earth.
αα
φθθ ddRd sin2
=∑

3. Apparent Magnitude
The flux at the Earth
∫ ∫∫∫ =
∑
≈Ω=
π π
θθθφ
θ
α
2
0
2/
02
2
2
sincos
cos
cos)( Idd
r
R
r
d
IdIrF
If the star has spherical symmetry, I(θθ)=I
2
4
2
2
2
2
4
)()(
r
L
T
r
R
RF
r
R
rF e
π
σ ===
2
1
12 log5.2
F
F
mm =−
m1 and m2 are the apparent magnitude of two stars, F1 and F2 are their fluxes at the Earth

4. •If the flux F is monochromatic at some wavelength, then m is a monochromatic
magnitude at that wavelength
• If the flux F is the integrated in all wavelengths, then m is a bolometric magnitude
•In general case, the measured magnitude depends on the efficiency of the equipment for
responding to radiation of wavelength λλ, ϕϕ(λλ), which has a value between 0 and 1 for all λλ.
The measured flux:
λλϕ λ drFFmeasured )()(
0∫
∞
=
•For ϕϕ(λλ) = 0 except in a very narrow wavelength range, the corresponding magnitude is
monochromatic
•A bolometric magnitude has ϕϕ(λλ) = 1 for all wavelengths
•The most common used one is Johnson’s system, U,B,V,R,I, J,H,K,L,M,N etc.
•Fλλ is the monochromatic flux measured at the Earth and corrected for the Earth’s
atmosphere

5. The response function for U,B,V, R and I bands
For a V band,
λλϕ λ drFF VV )()(
0∫
∞
=
VVV FCVm log5.2−==
Similar relations hold for other wavebands!
CV is the visible system constant

6. Black-body colors: Stellar energy distributions do appear rather similar to those of
Black bodies, at least in general form.
Since ϕϕ(λλ) is a rather narrow function,
V
FF VV λλ∆≈
where λλV is the average visual wavelength (about 5500 Å) and ∆∆ λλV is the average
width of ϕϕ(λλ) (about 900 Å).
VVV FCVm log5.2−== V
FCV V λlog5.2−′=
1
/2
)( /
52
−
= kThc
e
hc
TB λλ
λ
λλλ πB
r
R
RF
r
R
rF 2
2
2
2
)()( ==
1
/2
)(
52
2
2
−
= x
e
hc
r
R
rF
λπ
λ
where kThcx λ/=
R is the radius of the star, r is the distance to the Earth

7. r
R
eCV Vx
VV log5)1log(5.2log5.12 −−++′′= λ
A corresponding equation will hold for any other magnitude system.
Colors:
1
1
log5.2log5.12)(
−
−
++=− V
B
x
x
V
B
BV
e
e
CVB
λ
λ
1
1
log5.2log5.12)(
−
−
++=− B
U
x
x
B
U
UB
e
e
CBU
λ
λ
For the Sun, 5800)( =SunTe °K 10.0)( +=− BU 62.0)( +=−VB
In order to reproduce
this color 60.0+=BVC 10.0−=UBC
Since stars are not black body, the colors found will differ by some from the correct colors

8. Table 1.6.1. Black-body colors
•colors become more negative (or blue) as T rises
•Colors becomes less sensitive at the higher T
•These colors are useful only at those temperatures
for which a fairly fraction of energy is radiated
near the wavelengths of the magnitude systems
•At the highest T, one would obtained more useful
information by using magnitude systems with
wavelengths much shorter than UBV, same applies
to very low T
• (B-V)-T relation is a fairly good approximation
to that of the stars, (U-B)-T is not!
Black body
Main sequence
stars

9. Calibration of the Color-temperature relation
2
4
log5.2log5.2
r
L
CFCm bolbolbol
π
−=−=
The apparent bolometric magnitude of a star can be expressed as
or
4
log5.2log5 ebolbol T
r
R
Cm σ−−=
Determining the effective T can be reduced to the problem of measuring its mbol and its
angular size. However, it is difficult to measure mbol.
Several ways to measure angular size (R/r): Eclipsing binaries, stellar interferometer

10. Bolometric correction: bolmVBC −=
Bolometric correction/color relation
for main sequence stars
Effective temperature/color relation
for main sequence stars
• the color-temperature is close to the black-body relation in Table 1.6.1.
• the difference becomes considerable for very hot and very cool stars.

11. 1.7. Stellar Spectra and H-R Diagrams
Spectral classification
1. Draper classification: O, B, A, F,
G, K, and M
Each letter group except O is
divided into about 10 subdivisions,
indicated by numbers from 0-9.
The O group only runs from 5-9
Early type means close to O5, late
type means close to M9. Early A
type is one around A0-A2, while a
late A is about A7-A9.
The spectral sequence is
essentially a temperature
sequence

13. Approximate line strengths for different spectral type
16.56.8Cr24
15.67.4Mn25
16.27.9Fe26
17.17.9Co27
14.76.7V23
13.66.8Ti22
12.86.5Sc21
11.96.1Ca20
31.64.3K19
27.615.8Ar18
23.813.0Cl17
23.310.4S16
19.710.5P15
16.38.2Si14
18.86.0Al13
15.07.6Mg12
47.35.1Na11
41.021.6Ne10
35.017.4F9
35.113.6O8
29.614.5N7
24.411.3C6
25.28.3B5
18.29.3Be4
75.65.4Li3
54.424.6He2
13.6H1
II (eV)I (eV)elementZ
Stage of ionization
000,30~eT 2.6 eV
000,6~eT 0.5 eV

14. Black Body Radiation Curves at Different T
200 eV20 eV
•At T ~ 1000 K, photons have
energy less than ~ 20 eV to be
absorbed by metal species
•At T ~ 50,000K, a lot of
photons have energy between ~
20-200 eV to be absorbed by
He, H

15. 2. Morgan-Keenan (MK) classification:
•two-dimensional, one is the same as the old Draper classes, running from O5 to
M9 and containing nearly 70 discrete classes.
•The other dimension is luminosity classes. These are Ia, Ib, II, III, IV, and V.
•The MK system can be calibrated with temperature and electron pressure
•A spectral class is determined mainly by its average degree of ionization. Once
temperature or pressure varies with luminosity class, the other must also varies in
order to keep the average ionization approximately constant.
•The spectral sequence is primarily a temperature sequence, the luminosity
sequence primarily an electron-pressure sequence

16. Hertzsprung-Russell Diagram (H-R diagram)
•A star is represented by a point in H-R diagram.
•The abscissa is quantity related to Te of the star, such as log Te, (B-V), or spectral
type,
•the ordinate is measure of the luminosity such as log L, or absolute magnitude
T increase
Lincrease