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Chapter 15
Normal Stars
John S. Drilling and Arlo U. Landolt
15.1
15.1
Stellar Quantities and Interrelations . . . . . . . . . . . 381
15.2
Spectral Classification . . . . . . . . . . . . . . . . . . 383
15.3
Photometric Systems . . . . . . . . . . . . . . . . . . . 385
15.4
Stellar Atmospheres . . . . . . . . . . . . . . . . . . . . 393
15.5
Stellar Structure . . . . . . . . . . . . . . . . . . . . . . 395
STELLAR QUANTITIES AND INTERRELATIONS
M = mass (M = Sun’s mass).
R = radius (R = Sun’s radius).
L = luminosity (L = Sun’s luminosity) = total outflow of radiation (ergs s−1 ).
ρ̄ = mean density = M/( 43 π R 3 ).
Sp = spectral classification, which may be combined with a luminosity class.
m = apparent magnitude = −2.5 log apparent brightness. Typical subscripts: V = visual,
B = blue, U = ultraviolet, pg = photographic, pv = photovisual, bol = bolometric (total
radiation); in general, m λ = apparent magnitude of spectral region λ.
U , B, V = m U , m B , m V = apparent magnitudes in the UBV system.
M = absolute magnitude = apparent magnitude standardized to 10 pc without interstellar
absorption.
B − V = color index; (B − V )0 = intrinsic color index (i.e., no interstellar absorption); or, in
general a color index is the difference in the apparent magnitude as measured at two
different wavelengths.
BC = bolometric correction = m bol − V (always negative).
A = space absorption in magnitudes (usually visual).
381
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N ORMAL S TARS
m 0 = magnitude corrected for absorption = m − A.
E B−V = color excess = (B − V ) − (B − V )0 .
m − M = distance modulus = 5 log d − 5 + A.
m 0 − M = distance modulus corrected for absorption = 5 log d − 5, where d is distance in
parsecs (pc).
F = total radiant flux at stellar surface.
f = radiant flux for a star outside the Earth’s atmosphere.
4 ), where σ is Stefan’s constant.
Teff = stellar effective temperature (from F = σ Teff
vrot = equatorial rotational velocity.
g = surface gravity (cm s−2 ).
d = distance, usually in parsecs (pc).
π = parallax in seconds of arc ( ) = 1/d, with d in pc.
All the logarithms in this chapter are common logs with a base of 10.
15.1.1
Numerical Relations
M = m + 5 + 5 log π − A = m + 5 − 5 log d − A,
Mbol = −2.5 log L/L + 4.74,
where L = 3.845 × 1033 ergs s−1 , and +4.74 is the absolute bolometric magnitude of the Sun. The
bolometric correction is the difference between the absolute visual and absolute bolometric magnitude:
BC = Mbol − MV .
Bolometric luminosities, radii, and effective temperatures are related by
Mbol = 42.36 − 5 log R/R − 10 log Teff ,
where solar values of Mbol = 4.74 and Teff = 5777 K have been adopted.
log L = −3.147 + 2 log R + 4 log Teff ,
(m bol = 0) star ≡ 2.48 × 10−5 erg cm−2 s−1 outside the Earth’s atmosphere,
(Mbol = 0) star ≡ 2.97 × 1028 watts emitted radiation,
(MV = 0) star ≡ 2.45 × 1029 candela.
The zero age main sequence (ZAMS) is represented by [1]
log R/R = 0.640 log M/M + 0.011
log R/R = 0.917 log M/M − 0.020
(0.12 < log M/M < 1.3),
(−1.0 < log M/M < 0.12).
The mass–luminosity relation may be written [2, 3]:
log M/M = 0.48 − 0.105Mbol
for
−8 ≤ Mbol < 10.5,
or
log L/L = 3.8 log M/M + 0.08
for M > 0.2M .
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15.2 S PECTRAL C LASSIFICATION / 383
Another representation is [4]
log M/M = 0.46 − 0.10Mbol ,
log M/M = 0.75 − 0.145Mbol ,
Mbol < 7.5,
Mbol > 7.5.
The most reliable stellar masses are summarized in [5] and [6]; also, see the discussion in [2].
15.2
SPECTRAL CLASSIFICATION
We define normal stars to be those which can be classified on the MK system (specifically, [7, 8], and
more generally [9]), or which are classified as white dwarfs according to the system described in [10].
Table 15.1 gives these classes.
Table 15.1. MK spectral classes.
MK spectral class
O
B
A
F
G
K
M
Class characteristics
Hot stars with He II absorption
He I absorption; H developing later
Very strong H, decreasing later; Ca II increasing
Ca II stronger; H weaker; metals developing
Ca II strong; Fe and other metals strong; H weaker
Strong metallic lines; CH and CN bands developing
Very red; TiO bands developing strongly
The spectral classes are further subdivided into decimal subclasses (e.g., B0, B1, B2, etc.), although
not all subdivisions are used, and some classes are further subdivided (e.g., O9.5). Table 15.2 lists the
MK luminosity classes.
Table 15.2. MK luminosity classes.
MK luminosity class
Examples
I supergiants
II bright giants
III giants
IV subgiants
V dwarfs (main sequence)
B0 I
B5 II
G0 III
G5 IV
G0 V
The luminosity classes are further subdivided (e.g., Ia, Iab, Ib, etc.).
The MK classification is based on the appearance of pairs of spectral lines in the blue spectral
region at a spectral resolution of approximately 2 Å, as compared to standard stars [7, 8]. The main
line pairs are as shown in Table 15.3 and are illustrated in [11], [12], and [13].
Table 15.3. Line pairs for spectral classes and luminosity.
Class
Line pairs for class
Class
Line pairs for luminosity
O5 ⇔ O9
B0 ⇔ B1
B2 ⇔ B8
4471 He I/4541 He II
4552 Si III/4089 Si IV
4128−30 Si II/4121 He I
O9 ⇔ B3
B0 ⇔ B3
B1 ⇔ A5
4116−21 (Si IV, He I)/4144 He I
3995 N II/4009 He II
Balmer line wings
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N ORMAL S TARS
Table 15.3. (Continued.)
Class
Line pairs for class
Class
Line pairs for luminosity
B8 ⇔ A2
4471 He I/4481 Mg II
4026 He I/3934 Ca II
4030−34 Mn I/4128−32
4300 CH/4385
4300 (G band)/4340 Hγ
4045 Fe I/4101 Hδ
4226 Ca I/4340 Hγ
4144 Fe I/4101 Hδ
4226 Ca I/4325
4290/4300
A3 ⇔ F0
4416/4481 Mg II
F0 ⇔ F8
4172/4226 Ca I
F2 ⇔ K5
4045−63 Fe I/4077 Sr II
4226 Ca I/4077 Sr II
Discontinuity near 4215
4215/4260, Ca I increasing
A2 ⇔ F5
F2 ⇔ K
F5 ⇔ G5
G5 ⇔ K0
K0 ⇔ K5
G5 ⇔ M
K3 ⇔ M
Other characteristics sometimes included with MK types:
e = emission lines, e.g., Be;
f = certain O type emission line stars;
p = peculiar spectrum;
n = broad lines;
s = sharp lines;
k = interstellar lines present;
m = metallic line star.
Additional classes [2] are shown in Table 15.4.
Table 15.4. Additional spectral classes
Spectral class
S
R (or C)
N (or C)
15.2.1
Class characteristics
Strong bands of ZrO and YO, LaO, TiO
Strong bands of CN and CO instead of TiO in class M
Swan bands of C2 , Na I (D), Ca I 4227, for the rest
similar to R
White Dwarf Spectral Classification
The following information on white dwarf spectral classification was provided by J. Liebert and
E. Sion ([10] and illustrated examples in [14]). The system consists of: (1) first symbol: an uppercase
D for a degenerate star spectrum; (2) second symbol: an uppercase letter designating the primary or
dominant ion or type of element in the optical spectrum; (3) third and possible subsequent symbols:
(optional) uppercase letters designating any secondary ions or types of elements appearing in the
optical spectrum, usually due to species with trace abundances (special secondary symbols are also
provided for spectra showing polarized light and magnetic fields, and others with peculiar spectra);
and (4) a temperature index defined by 10θeff , which is equal to 50 400/Teff . Originally, this index was
specified to be a single digit from 0 to 9. This index can be estimated only from at least a rough analysis
of spectrophotometric data, using colors, an energy distribution or the strengths of absorption features.
In this way, the system differs from traditional, purely spectroscopic methodology. If such information
is unavailable or ambiguous, the temperature subtype is omitted.
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15.3 P HOTOMETRIC S YSTEMS / 385
DA
DB
DO
DZ
DQ
DX, DXP
Definition of Primary Symbols
Hydrogen Balmer lines dominate optical spectrum.
Neutral helium (He I) lines dominate.
Ionized helium lines strongest, He I and/or H may be visible.
Metal lines dominate, usually with Ca II strongest.
Carbon features, either molecular or atomic, in any part of the electromagnetic
spectrum (often strongest in the ultraviolet).
Star with unidentified features, presumably due to a strong magnetic field. If light
polarized, the secondary symbol “P” is also appropriate.
Secondary Symbols: All of the Above, Plus . . .
Star showing polarized light.
Star known to be magnetic from optical Zeeman features, but not known to be
polarized.
Star known to be photometrically variable (optional).
Star with spectral peculiarities.
P
H
V
PEC
Examples
DA1
DAO1
DOZ1
DBAQ4
DXP5
DZA7
DC9
15.3
A white dwarf showing only hydrogen lines with 37 500 < Teff < 100 000 K.
Star in same temperature range showing hydrogen and weak He II.
A star showing strong He II, weak He I, H, and N V features at Teff = 70 000 K.
A star showing He I, H, and C features in that order of decreasing strengths, near
Teff = 12 000 K.
A polarized, magnetic white dwarf with unidentified spectral features,
Teff ∼ 10 000 K.
A metallic line white dwarf also showing weak hydrogen lines, Teff = 8500 K.
A featureless, continuous spectrum with an estimated Teff = 5500 K.
PHOTOMETRIC SYSTEMS
Various photometric systems are used to supplement or replace the spectral classifications referred to
in the last section. Optical filters are used to isolate specific spectral features or wavelength ranges, and
the fluxes received through these filters are usually expressed in magnitudes,
m = −2.5 log( f / f 0 ),
where f is the measured flux (corrected for atmospheric effects), and f 0 is the corresponding flux for
a star with m = 0. The system is defined by the magnitudes and color indices (magnitude differences)
for a set of standard stars, which have been determined using a particular instrumental setup. The
standard stars are used to transform measurements made with other instrumental setups to the standard
system. Also important for theoretical studies are the sensitivity functions (response of the original
instrumental setup to a source that emits the same flux at all wavelengths) for the various filters as a
function of wavelength. The effective wavelengths (peak sensitivity) and widths at half maximum of
the sensitivity functions for selected photometric systems in common use at the present time are given
in Table 15.5. References containing lists of standard stars, sensitivity functions, and calibrations, are
indicated in the last column.
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N ORMAL S TARS
Table 15.5. Modern photometric systems.
System
Strömgren four-color system
Geneva seven-color system
Vilnius seven-color system
Walraven system
Washington system
DDO five-color system
RGU
UBVRI and (RI)KC
Characteristic wavelength passbands
(effective wavelengths and half-widths) (Å)
3500 (380), 4100 (200), 4700 (200), 5550
(200), plus Hβ (150/30)
UBV system plus 4020 (170), 4480 (165),
5400 (200), 5810 (210)
3450 (400), 3740 (260), 4050 (220), 4660
(260), 5160 (210), 5440 (260), 6550 (200)
5400 (710), 4300 (540), 3820 (430), 3620
(230), 3250 (140)
3910 (1100), 5085 (1050), 6330 (800), 8050
(1500)
4886 (186), 4517 (76), 4257 (73), 4166 (83),
additional: 3815 (330), 3460 (383)
3593 (530), 4658 (495), 6407 (430)
3600 (700), 4400 (1000), 5500 (900), 7000
(2200), 8800 (2400), 6400 (1750), 7900
(1400)
Designations
References
uvbyβ
[1–6]
U BV B1 B2 V1 G
[1, 7–9]
[10–13]
[1, 14–16]
UPXYZVS
VBLUW
CMT1 T2
C(41–42)
C(42–45)
C(45–48)
RGU
UBVRI
[1, 7, 17, 18]
[19, 20]
[21–26]
[1, 7, 27, 28]
[1, 7, 9, 29–33]
[1, 34–38]
References
1. Schmidt-Kaler, Th. 1982, Landolt-Börnstein: Numerical Data and Functional Relationships in Science and
Technology, edited by K. Schaifers and H.H. Voigt (Springer-Verlag, Berlin), VI/2b
2. Crawford, D.L. 1975, AJ, 80, 955
3. Crawford, D.L. 1978, AJ, 83, 48
4. Crawford, D.L. 1979, AJ, 84, 1858
5. Olson, E.C. 1974, PASP, 86, 80
6. Strömgren, B. 1966, ARA&A, 4, 433
7. Golay, M. 1974, Introduction to Astronomical Photometry (Reidel, Dordrecht)
8. Rufener, F., & Maeder, A. 1971, A&AS, 4, 43
9. Philip, A.G.D., editor, 1979, Problems of Calibration of Multicolor Photometric Systems (Davis, Schenectady)
10. Hauck, B. 1985, in Calibration of Fundamental Stellar Quantities, edited by D.S. Hayes, L.E. Pasinetti, and A.G.D.
Philip (Kluwer Academic), p. 271
11. North, P., & Nicolet, B. 1990, A&A, 228, 78
12. Rufener, F., & Nicolet, B. 1988, A&A, 206, 357
13. Meynet, G., & Hauck, B. 1985, A&A, 150, 163
14. Straizys, V., & Zdanavicius, K. 1970, Bull. Vilnius Astron. Obs. No. 29, 15
15. Straizys, V., 1977, Multicolor Stellar Photometry, Photometric Systems and Methods (Mokslos, Vilnius)
16. Straizys, V., & Jodinskiene, E. 1981, Bull. Vilnius Astron. Obs. No. 56
17. Lub, J., & Pel, J.W. 1977, A&A, 54, 137
18. Pel, J.W. 1976, A&AS, 24, 413
19. De Ruiter, H.R., & Lub, J. 1986, A&AS, 63, 59
20. Brand, J., & Wouterloot, J.G.A. 1988, A&AS, 75, 117
21. Canterna, R. 1976, AJ, 81, 228
22. Canterna, R., & Harris, H.C. 1979, Dudley Obs. Rep. No. 14; op. cit. [9], p. 199
23. Harris, H.C., & Canterna, R. 1979, AJ, 84, 1750
24. Geisler, D. 1984, PASP, 96, 723
25. Geisler, D. 1990, PASP, 102, 344
26. Geisler, D., Claria, J.J., & Minniti, D. 1991, AJ, 102, 1836
27. McClure, R.D. 1976, AJ, 81, 182
28. McClure, R.D., & van den Bergh, S. 1968, AJ, 73, 313
29. Steinlin, U.W. 1968, Z. Astrophys., 69, 276
30. Smith, L.L., & Steinlin, U.W. 1964, Z. Astrophys., 58, 253
31. Bell, R.A. 1972, MNRAS, 159, 34; 1972, A&A, 62, 411
32. Buser, R. 1978, A&A, 62, 411
33. Buser, R. 1978, A&A, 62, 425
34. Cousins, A.W.J. 1976, MemRAS, 81, 25
35. Landolt, A.U. 1992, AJ, 104, 340
36. Bessell, M.S. 1979, PASP, 91, 589
Sp.-V/AQuan/1999/10/10:10:18
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15.3 P HOTOMETRIC S YSTEMS / 387
37. Bessell, M.S. 1976, PASP, 88, 557
38. Menzies, J.W. et al. 1991, MNRAS, 248, 642
Absolute calibration of a star of the spectral type A0 V with the magnitude V = 0 [2] on the Johnson
system is shown in Table 15.6.
Table 15.6. Flux calibration for an A0 V star.
Symbol
Flux (erg cm−2 s−1 Å−1 )
λ0 (µm)
U
B
V
R
I
4.22 × 10−9
0.36
0.44
0.55
0.71
0.97
6.40 × 10−9
3.75 × 10−9
1.75 × 10−9
8.4 × 10−10
Useful relations for the UBV system [2]:
(U − B)0 = 0.08 + 3.85(B − V )0 unreddened main sequence, (B − V )0 < 0 and (U − B)0 < 0,
Q = (U − B) − 0.72(B − V ) independent of reddening for early-type stars,
EU −B
(U − B)0 < 0,
0.65 − 0.05(U − B)0 + 0.05E B−V ,
=
(B − V )0 > 0,
0.64 + 0.26(B − V )0 + 0.05E B−V ,
E B−V
AV
= 3.30 + 0.28(B − V )0 + 0.04E B−V ,
E B−V
where EU −B = (U − B) − (U − B)0 , A V = V − V0 , E B−V = (B − V ) − (B − V )0 ; and V0 , (B − V )0 ,
and (U − B)0 are the magnitude and color indices stars would have if space were transparent.
Useful relations for the uvbyβ system [15–20]:
c1 = (u − v) − (v − b),
m 1 = (v − b) − (b − y),
β = 2.5 log(W/N ),
where W and N are the fluxes measured through interference filters centered on Hβ with half-widths
of about 150 and 30 Å, respectively.

E(c1 ) = 0.20E(b − y), 
E(m 1 ) = −0.32E(b − y), color excesses according to standard reddening law,

E(u − b) = 1.50E(b − y),

[c1 ] = c1 − 0.20(b − y),

[m 1 ] = m 1 + 0.32(b − y),
reddening independent quantities,

[u − b] = (u − b) − 1.50(b − y),
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N ORMAL S TARS
(b − y)0 = −0.116 + 0.097c1 for an unreddened main-sequence B star,
(b − y)0 = 2.946 − 1.0β − 0.1δc1 (−0.25δm 1 if m 1 < 0) for A stars with
2.870 > β > 2.720 and δc1 < 0.28,
(b − y)0 = 0.222 + 1.11β + 2.7(β)2 − 0.05δc1 − (0.1 + 3.6β)δm 1 for F stars
with 2.630 < β < 2.720 and δc1 < 0.28, or 2.590 < β < 2.630 and
δc1 < 0.20,
where β = 2.720 − β, δc1 = c1 − cstd , δm 1 = m std − m 1 ; See Section 15.3.2 for cstd and m std .
15.3.1
Calibration of MK Spectral Types [2, 21, 22]
Table 15.7 presents the absolute magnitude, color, effective surface temperature, and bolometric
correction calibrations for the MK spectral classes. Table 15.8 gives the calibrated physical parameters
for stars of the various spectral classes.
Table 15.7. Calibration of MK spectral types.
B−V
U−B
V−R
R−I
Teff
BC
MAIN SEQUENCE, V
O5
−5.7
−0.33
O9
−4.5
−0.31
B0
−4.0
−0.30
B2
−2.45
−0.24
B5
−1.2
−0.17
B8
−0.25
−0.11
A0
+0.65
−0.02
A2
+1.3
+0.05
A5
+1.95
+0.15
F0
+2.7
+0.30
F2
+3.6
+0.35
F5
+3.5
+0.44
F8
+4.0
+0.52
G0
+4.4
+0.58
G2
+4.7
+0.63
G5
+5.1
+0.68
G8
+5.5
+0.74
K0
+5.9
+0.81
K2
+6.4
+0.91
K5
+7.35
+1.15
M0
+8.8
+1.40
M2
+9.9
+1.49
M5
+12.3
+1.64
−1.19
−1.12
−1.08
−0.84
−0.58
−0.34
−0.02
+0.05
+0.10
+0.03
0.00
−0.02
+0.02
+0.06
+0.12
+0.20
+0.30
+0.45
+0.64
+1.08
+1.22
+1.18
+1.24
−0.15
−0.15
−0.13
−0.10
−0.06
−0.02
0.02
0.08
0.16
0.30
0.35
0.40
0.47
0.50
0.53
0.54
0.58
0.64
0.74
0.99
1.28
1.50
1.80
−0.32
−0.32
−0.29
−0.22
−0.16
−0.10
−0.02
0.01
0.06
0.17
0.20
0.24
0.29
0.31
0.33
0.35
0.38
0.42
0.48
0.63
0.91
1.19
1.67
42 000
34 000
30 000
20 900
15 200
11 400
9 790
9 000
8 180
7 300
7 000
6 650
6 250
5 940
5 790
5 560
5 310
5 150
4 830
4 410
3 840
3 520
3 170
−4.40
−3.33
−3.16
−2.35
−1.46
−0.80
−0.30
−0.20
−0.15
−0.09
−0.11
−0.14
−0.16
−0.18
−0.20
−0.21
−0.40
−0.31
−0.42
−0.72
−1.38
−1.89
−2.73
GIANTS, III
G5
+0.9
G8
+0.8
K0
+0.7
K2
+0.5
K5
−0.2
M0
−0.4
M2
−0.6
M5
−0.3
+0.56
+0.70
+0.84
+1.16
+1.81
+1.87
+1.89
+1.58
0.69
0.70
0.77
0.84
1.20
1.23
1.34
2.18
0.48
0.48
0.53
0.58
0.90
0.94
1.10
1.96
5 050
4 800
4 660
4 390
4 050
3 690
3 540
3 380
−0.34
−0.42
−0.50
−0.61
−1.02
−1.25
−1.62
−2.48
Sp
M(V )
+0.86
+0.94
+1.00
+1.16
+1.50
+1.56
+1.60
+1.63
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15.3 P HOTOMETRIC S YSTEMS / 389
Table 15.7. (Continued.)
Sp
M(V )
B−V
SUPERGIANTS, I
O9
−6.5
−0.27
B2
−6.4
−0.17
B5
−6.2
−0.10
B8
−6.2
−0.03
A0
−6.3
−0.01
A2
−6.5
+0.03
A5
−6.6
+0.09
F0
−6.6
+0.17
F2
−6.6
+0.23
F5
−6.6
+0.32
F8
−6.5
+0.56
G0
−6.4
+0.76
G2
−6.3
+0.87
G5
−6.2
+1.02
G8
−6.1
+1.14
K0
−6.0
+1.25
K2
−5.9
+1.36
K5
−5.8
+1.60
M0
−5.6
+1.67
M2
−5.6
+1.71
M5
−5.6
+1.80
U−B
V−R
R−I
Teff
BC
−1.13
−0.93
−0.72
−0.55
−0.38
−0.25
−0.08
+0.15
+0.18
+0.27
+0.41
+0.52
+0.63
+0.83
+1.07
+1.17
+1.32
+1.80
+1.90
+1.95
+1.60:
−0.15
−0.05
0.02
0.02
0.03
0.07
0.12
0.21
0.26
0.35
0.45
0.51
0.58
0.67
0.69
0.76
0.85
1.20
1.23
1.34
2.18
−0.32
−0.15
−0.07
0.00
0.05
0.07
0.13
0.20
0.21
0.23
0.27
0.33
0.40
0.44
0.46
0.48
0.55
0.90
0.94
1.10
1.96
32 000
17 600
13 600
11 100
9 980
9 380
8 610
7 460
7 030
6 370
5 750
5 370
5 190
4 930
4 700
4 550
4 310
3 990
3 620
3 370
2 880
−3.18
−1.58
−0.95
−0.66
−0.41
−0.28
−0.13
−0.01
−0.00
−0.03
−0.09
−0.15
−0.21
−0.33
−0.42
−0.50
−0.61
−1.01
−1.29
−1.62
−3.47
Table 15.8. Calibration of MK spectral types.a
Sp
M/M
R/R
MAIN SEQUENCE, V
O3
120
15
O5
60
12
O6
37
10
O8
23
8.5
B0
17.5
7.4
B3
7.6
4.8
B5
5.9
3.9
B8
3.8
3.0
A0
2.9
2.4
A5
2.0
1.7
F0
1.6
1.5
F5
1.4
1.3
G0
1.05
1.1
G5
0.92
0.92
K0
0.79
0.85
K5
0.67
0.72
M0
0.51
0.60
M2
0.40
0.50
M5
0.21
0.27
M8
0.06
0.10
log(g/g )
log(ρ̄/ρ̄ )
vrot (km s−1 )
−0.3
−0.4
−0.45
−0.5
−0.5
−0.5
−0.4
−0.4
−0.3
−0.15
−0.1
−0.1
−0.05
+0.05
+0.05
+0.1
+0.15
+0.2
+0.5
+0.5
−1.5
−1.5
−1.45
−1.4
−1.4
−1.15
−1.00
−0.85
−0.7
−0.4
−0.3
−0.2
−0.1
−0.1
+0.1
+0.25
+0.35
+0.8
+1.0
+1.2
200
170
190
240
220
180
170
100
30
10
< 10
< 10
< 10
Sp.-V/AQuan/1999/10/10:10:18
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390 / 15
N ORMAL S TARS
Table 15.8. (Continued.)
R/R
log(g/g )
log(ρ̄/ρ̄ )
vrot (km s−1 )
GIANTS, III
B0
20
B5
7
A0
4
G0
1.0
G5
1.1
K0
1.1
K5
1.2
M0
1.2
15
8
5
6
10
15
25
40
−1.1
−0.95
−1.5
−1.9
−2.3
−2.7
−3.1
−2.2
−1.8
−1.5
−2.4
−3.0
−3.5
−4.1
−4.7
120
130
100
30
< 20
< 20
< 20
SUPERGIANTS, I
O5
70
O6
40
O8
28
B0
25
B5
20
A0
16
A5
13
F0
12
F5
10
G0
10
G5
12
K0
13
K5
13
M0
13
M2
19
30:
25:
20
30
50
60
60
80
100
120
150
200
400
500
800
−1.1
−1.2
−1.2
−1.6
−2.0
−2.3
−2.4
−2.7
−3.0
−3.1
−3.3
−3.5
−4.1
−4.3
−4.5
−2.6
−2.6
−2.5
−3.0
−3.8
−4.1
−4.2
−4.6
−5.0
−5.2
−5.3
−5.8
−6.7
−7.0
−7.4
125
102
40
40
38
30
< 25
< 25
< 25
< 25
< 25
Sp
M/M
Note
a A colon indicates an uncertain value.
Also see [23]. An independent absolute magnitude calibration is given in graphical form in [8]. Plots
of (B − V ) and (U − V ) versus MV for the various white dwarf subclasses are in [24]. Intrinsic colors
and absolute magnitudes of the zero-age main sequence (ZAMS) (locus of young stars just starting
hydrogen burning) follow [2]. See [25] for an alternative, and plots in Chapter 20. Table 15.9 gives the
zero-age main sequence colors and absolute magnitudes.
Table 15.9. Zero-age main sequence.
(B − V )0
(U − B)0
−0.m 33
−0.305
−0.30
−0.28
−0.25
−0.22
−0.20
−0.15
−0.10
−0.05
0.00
+0.05
+0.10
−1.m 20
−1.10
−1.08
−1.00
−0.90
−0.80
−0.69
−0.50
−0.30
−0.10
+0.01
+0.05
+0.08
Mv
−5.m 2
−3.6
−3.25
−2.6
−2.1
−1.5
−1.1
−0.2
+0.6
+1.1
+1.5
+1.7
+1.9
(B − V )0
(U − B)0
+0.40
+0.50
+0.60
+0.70
+0.80
+0.90
+1.00
+1.10
+1.20
+1.30
+1.40
+1.50
+1.60
−0.01
0.00
+0.08
+0.23
+0.42
+0.63
+0.86
+1.03
+1.13
+1.20
+1.22
+1.17
+1.20
Mv
+ 3.4
+ 4.1
+ 4.7
+ 5.2
+ 5.8
+ 6.3
+ 6.7
+ 7.1
+ 7.5
+ 8.0
+ 8.8
+10.3
+12.0
Sp.-V/AQuan/1999/10/10:10:18
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15.3 P HOTOMETRIC S YSTEMS / 391
Table 15.9. (Continued.)
15.3.2
(B − V )0
(U − B)0
+0.15
+0.20
+0.25
+0.30
+0.35
+0.09
+0.10
+0.07
+0.03
0.00
Mv
+2.1
+2.4
+2.55
+2.8
+3.1
(B − V )0
(U − B)0
+1.70
+1.80
+1.90
+2.00
+1.32
+1.43
+1.53
+1.64
Mv
+13.2
+14.2
+15.5
+16.7
u v b y β Standard Relations
For the early-type stars, Table 15.10 gives the standard relation between the β index, colors, and the
absolute magnitudes.
Table 15.10. uvbyβ standard relations.
β
b−y
m1
c1
2.590
2.600
2.620
2.640
2.660
2.680
2.700
2.720
2.740
2.760
2.780
2.800
2.820
2.840
2.860
2.880
2.900
2.910
−0.134
−0.126
−0.118
−0.109
−0.100
−0.091
−0.080
−0.070
−0.061
−0.050
−0.044
−0.041
−0.039
−0.037
−0.034
−0.029
−0.023
−0.020
0.045
0.055
0.075
0.080
0.085
0.093
0.100
0.100
0.109
0.110
0.116
0.120
0.120
0.123
0.128
0.132
0.138
0.140
−0.250
−0.128
−0.025
0.065
0.150
0.235
0.321
0.404
0.491
0.578
0.656
0.724
0.785
0.833
0.878
0.925
0.975
1.000
2.880
2.870
2.860
2.850
2.840
2.830
2.820
2.810
2.800
2.790
2.780
2.770
2.760
2.750
2.740
2.730
2.720
0.066
0.076
0.086
0.096
0.106
0.116
0.126
0.136
0.146
0.156
0.166
0.176
0.186
0.196
0.206
0.216
0.226
0.200
0.202
0.205
0.206
0.208
0.207
0.206
0.204
0.203
0.200
0.196
0.192
0.188
0.185
0.182
0.180
0.177
MV
[m 1 ]
[c1 ]
−4.65
−4.12
−3.17
−2.36
−1.69
−1.12
−0.65
−0.27
0.04
0.30
0.51
0.68
0.83
0.97
1.10
1.24
1.39
1.46
0.005
0.017
0.040
0.047
0.055
0.066
0.076
0.079
0.091
0.095
0.103
0.108
0.108
0.112
0.118
0.123
0.131
0.134
−0.223
−0.103
−0.001
0.087
0.170
0.253
0.337
0.418
0.503
0.588
0.665
0.732
0.793
0.840
0.885
0.931
0.980
1.004
2.30
2.40
2.50
2.57
2.64
2.67
2.70
2.73
2.76
2.79
2.82
2.85
2.88
2.92
2.96
3.03
3.10
0.220
0.225
0.231
0.235
0.240
0.242
0.244
0.245
0.247
0.247
0.246
0.245
0.244
0.244
0.244
0.245
0.245
0.917
0.895
0.873
0.851
0.829
0.812
0.795
0.773
0.751
0.729
0.707
0.685
0.663
0.641
0.619
0.587
0.555
B Stars
A Stars
0.930
0.910
0.890
0.870
0.850
0.835
0.820
0.800
0.780
0.760
0.740
0.720
0.700
0.680
0.660
0.630
0.600
Sp.-V/AQuan/1999/10/10:10:18
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392 / 15
N ORMAL S TARS
Table 15.10. (Continued.)
β
b−y
m1
c1
MV
[m 1 ]
[c1 ]
3.14
3.21
3.29
3.38
3.48
3.60
3.74
3.88
4.04
4.20
4.36
4.52
4.70
4.90
0.244
0.244
0.246
0.248
0.251
0.256
0.263
0.272
0.281
0.292
0.304
0.317
0.332
0.350
0.536
0.513
0.481
0.443
0.411
0.383
0.355
0.327
0.304
0.281
0.258
0.235
0.211
0.188
F Stars
2.720
2.710
2.700
2.690
2.680
2.670
2.660
2.650
2.640
2.630
2.620
2.610
2.600
2.590
0.222
0.233
0.245
0.258
0.271
0.284
0.298
0.313
0.328
0.344
0.360
0.377
0.394
0.412
0.177
0.174
0.172
0.171
0.170
0.171
0.174
0.178
0.183
0.189
0.196
0.204
0.214
0.226
0.580
0.560
0.530
0.495
0.465
0.440
0.415
0.390
0.370
0.350
0.330
0.310
0.290
0.270
See [15–17] and [26]. See also [27] and [28] for grids for determining effective temperatures and
surface gravities. Other calibrations may be found in [29–38].
15.3.3
Empirical U B V ( R I ) KC Calibrations [39]
The colors and spectral classes are given as a function of the surface effective temperature for dwarf
and giant stars in Table 15.11.
Table 15.11. Empirical U BV (R I )KC calibrations.
Teff
b−y
B−V
13 000
12 000
11 000
10 000
9 500
9 000
8 500
8 000
7 500
7 000
6 500
6 000
5 500
5 000
4 500
4 000
3 500
3 000
2 750
−0.054
−0.041
−0.027
−0.010
+0.007
+0.035
+0.072
+0.118
+0.165
+0.220
+0.286
+0.360
+0.445
+0.535
+0.60
+0.80
+1.01
+1.22
+1.37
−0.14
−0.10
−0.065
−0.025
+0.005
+0.055
+0.14
+0.22
+0.275
+0.35
+0.45
+0.57
+0.70
+0.88
+1.02
+1.32
+1.53
+1.74
+2.0
(V − R)KC
(R − I )KC
(V − I )KC
MK
−0.070
−0.050
−0.032
−0.012
+0.008
+0.040
+0.084
+0.132
+0.168
+0.207
+0.250
+0.303
+0.364
+0.43
+0.51
+0.74
+1.18
+1.77
+2.18
−0.120
−0.085
−0.055
−0.020
+0.015
+0.072
+0.155
+0.250
+0.330
+0.415
+0.515
+0.625
+0.760
+0.93
+1.11
+1.53
+2.19
+3.03
+3.58
B7
B8
B9
A0
A1
A2
A5
A7
F0
F2
F5
G0
G6
K2
K4
K7
M2
M4.5
M6
Dwarfs (V)
−0.050
−0.035
−0.023
−0.008
+0.007
+0.032
+0.071
+0.118
+0.162
+0.208
+0.265
+0.322
+0.396
+0.50
+0.60
+0.79
+1.01
+1.26
+1.40
Sp.-V/AQuan/1999/10/10:10:18
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15.4 S TELLAR ATMOSPHERES / 393
Table 15.11. (Continued.)
Teff
b−y
B−V
5 000
4 750
4 500
4 250
4 000
3 750
3 500
3 250
+0.55
+0.60
+0.68
+0.80
+0.90
+1.00
+0.89
+0.98
+1.11
+1.26
+1.43
+1.62
(V − R)KC
(R − I )KC
(V − I )KC
MK
Giants (III)
15.4
+0.497
+0.539
+0.60
+0.68
+0.795
+0.945
+1.19
+0.433
+0.461
+0.510
+0.600
+0.735
+1.025
+1.57
+0.93
+1.00
+1.11
+1.28
+1.53
+1.97
+2.76
+3.80
G7
K0
K2
K3
K5
M2
M4.5
M6
STELLAR ATMOSPHERES
15.4.1
Model Atmospheres for Normal Stars (Solar Composition) [40]
Table 15.12 lists stellar atmosphere parameters depending on the surface effective temperature and
gravity of a star.
Table 15.12. Model atmospheres for normal stars.
Teff
log g
log τ a
log x
T
log P
log n e
log n a
log ρ
log Pr
Fconv
F
5 500
4
−3.0
−2.0
−1.0
0.0
1.0
6.79
7.65
7.92
8.08
8.14
4 282
4 487
4 846
6 130
8 176
3.23
3.84
4.41
4.92
5.10
11.35
11.91
12.49
13.50
14.94
15.47
16.05
16.59
16.99
17.04
−8.19
−7.61
−7.07
−6.66
−6.62
0.09
0.10
0.17
0.54
1.05
0.00
0.00
0.00
0.01
0.85
5 500
1
−3.0
−2.0
−1.0
0.0
1.0
10.65
10.98
11.14
11.22
11.24
4 104
4 444
4 846
6 145
8 431
1.28
2.09
2.73
3.13
3.18
9.53
10.35
11.08
12.55
14.06
13.52
14.30
14.91
15.20
15.07
−10.13
−9.36
−8.75
−8.46
−8.58
0.09
0.10
0.17
0.56
1.10
0.00
0.00
0.00
0.00
0.91
6 000
4
−3.0
−2.0
−1.0
0.0
1.0
7.60
7.90
8.08
8.18
8.22
4 667
4 891
5 293
6 789
8 709
3.29
3.87
4.42
4.82
4.95
11.48
12.04
12.62
13.94
15.12
15.49
16.04
16.55
16.85
16.86
−8.17
−7.61
−7.10
−6.81
−6.79
0.24
0.25
0.32
0.70
1.16
0.00
0.00
0.00
0.05
0.88
6 000
1
−3.0
−2.0
−1.0
0.0
1.0
10.75
11.03
11.17
11.24
11.25
4 489
4 869
5 318
6 861
8 981
1.26
2.02
2.59
2.89
2.92
9.72
10.62
11.44
13.01
14.11
13.47
14.19
14.72
14.90
14.73
−10.19
−9.47
−8.94
−8.75
−8.93
0.24
0.25
0.33
0.75
1.21
0.00
0.00
0.00
0.00
0.91
7 000
4
−3.0
−2.0
−1.0
0.0
1.0
7.63
7.95
8.12
8.20
8.24
5 458
5 726
6 190
8 217
9 911
3.10
3.67
4.17
4.45
4.55
11.87
12.45
13.13
14.63
15.37
15.22
15.77
16.23
16.39
16.37
−8.44
−7.89
−7.42
−7.26
−7.28
0.51
0.52
0.60
1.02
1.38
0.00
0.00
0.00
0.20
0.92
Sp.-V/AQuan/1999/10/10:10:18
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394 / 15
N ORMAL S TARS
Table 15.12. (Continued.)
Teff
log g
log τ a
log x
T
log P
log n e
log n a
log ρ
log Pr
Fconv
F
10 000
4
−3.0
−2.0
−1.0
0.0
1.0
8.34
8.48
8.58
8.65
8.83
7 586
8 030
8 982
11 655
16 287
1.71
2.36
2.86
3.17
3.75
12.84
13.42
14.08
14.62
15.08
13.63
14.26
14.67
14.71
15.12
−10.03
−9.40
−8.99
−8.95
−8.54
1.13
1.15
1.28
1.68
2.25
0.00
0.00
0.00
0.00
0.00
20 000
4
−3.0
−2.0
−1.0
0.0
1.0
8.70
8.90
9.02
9.15
9.28
13 060
14 067
15 560
19 521
27 451
1.38
2.09
2.71
3.33
4.03
12.81
13.49
14.07
14.60
15.15
12.84
13.52
14.08
14.60
15.15
−10.82
−10.14
−9.57
−9.05
−8.50
2.34
2.35
2.40
2.63
3.15
0.00
0.00
0.00
0.00
0.00
40 000
4
−3.0
−2.0
−1.0
0.0
1.0
9.48
9.66
9.77
9.87
9.97
28 059
31 336
34 855
40 920
53 682
1.19
2.16
2.93
3.55
4.21
12.31
13.24
13.96
14.52
15.06
12.29
13.21
13.93
14.48
15.02
−11.37
−10.45
−9.72
−9.18
−8.64
3.54
3.55
3.62
3.85
4.32
0.00
0.00
0.00
0.00
0.00
Note
a τ = continuum optical depth (5000 Å); x = geometric depth; T = temperature (K); P = pressure; n = electron
e
number density; n a = atom number density; ρ = mass density; Pr = radiation pressure; Fconv /F = fraction of flux
carried by convection. All units are cgs.
Model atmospheres for metal-deficient stars are given in [40] and [41].
15.4.2
Theoretical Physical Continuum Fluxes [40]
Logarithms of theoretical physical continuum fluxes (ergs cm−2 s−1 Å−1 ) for normal stars (solar
composition) with log g = 4 [40] are given in Table 15.13.
Table 15.13. Continuum fluxes for normal stars.
λ (Å)
Teff (K) = 5500
506
890
920
1 482
2 012
2 506
3 012
3 636
3 661
4 012
4 512
5 025
5 525
6 025
7 075
8 152
8 252
10 050
14 594
−∞
−∞
−5.80
0.05
4.14
5.91
6.80
6.86
6.94
6.94
6.92
6.89
6.86
6.81
6.72
6.62
6.61
6.45
6.13
6000
7000
−∞
−∞
−4.49
1.63
5.41
6.55
7.00
7.04
7.16
7.15
7.12
7.08
7.03
6.97
6.86
6.75
6.74
6.56
6.19
−∞
−∞
−2.07
3.83
6.88
7.19
7.29
7.27
7.56
7.52
7.46
7.38
7.31
7.24
7.09
6.95
6.95
6.74
6.29
10 000
−6.26
1.11
3.73
8.28
8.12
7.98
7.89
7.79
8.33
8.21
8.06
7.92
7.79
7.68
7.46
7.26
7.33
7.03
6.47
20 000
40 000
4.81
7.34
10.28
9.84
9.48
9.22
8.99
8.76
8.93
8.80
8.63
8.46
8.32
8.19
7.94
7.71
7.73
7.41
6.80
11.19
10.93
11.28
10.75
10.35
10.05
9.78
9.49
9.49
9.35
9.17
8.99
8.84
8.70
8.44
8.20
8.19
7.86
7.23
Sp.-V/AQuan/1999/10/10:10:18
Page 395
15.5 S TELLAR S TRUCTURE / 395
Table 15.13. (Continued.)
λ (Å)
27 000
50 000
100 000
200 000
15.5
Teff (K) = 5500
5.22
4.20
3.03
1.83
6000
7000
10 000
5.27
4.24
3.06
1.87
5.34
4.31
3.13
1.94
5.48
4.45
3.27
2.07
20 000
40 000
5.77
4.72
3.52
2.32
6.17
5.10
3.89
2.67
STELLAR STRUCTURE
Age-zero models for X = 0.70, Z = 0.02, l/H p = 1.7 [1]. l = mixing length; Pc = central pressure
(dyn cm−2 ); Tc = central temperature (K); ρc = central density (g cm−3 ); H p = pressure scale
height; qcc = fraction of stellar mass within convective core; qce = fraction of stellar mass at bottom
of convective envelope are given in Table 15.14.
Table 15.14. Age-zero models [1].
M/M
10.0
9.0
8.0
7.0
6.0
5.0
4.0
3.0
2.5
2.0
1.8
1.6
1.4
1.2
1.0
0.8
log L/L 3.7600
3.6118
3.4433
3.2480
3.0184
2.7381
2.3815
1.9087
1.6002
1.2157
1.0298
0.8186
0.5562
0.2325
−0.1523
−0.5742
log Teff
log g
R/R
log Pc
log Tc
log ρc
qcc
qce
4.4096
4.3855
4.3576
4.3251
4.2865
4.2380
4.1766
4.0928
4.0376
3.9658
3.9297
3.8855
3.8355
3.7964
3.7514
3.7016
4.269
4.275
4.280
4.287
4.296
4.303
4.317
4.330
4.338
4.339
4.334
4.317
4.322
4.422
4.548
4.674
3.8434
3.6213
3.3916
3.1472
2.8860
2.6122
2.2992
1.9620
1.7737
1.5858
1.5120
1.4532
1.3526
1.1157
0.8813
0.6820
16.6331
16.6688
16.7116
16.7617
16.8243
16.8995
16.9982
17.1280
17.2085
17.2943
17.3260
17.3476
17.3274
17.2645
17.1851
17.0651
7.5051
7.4959
7.4853
7.4729
7.4583
7.4400
7.4163
7.3844
7.3631
7.3333
7.3151
7.2913
7.2500
7.1936
7.1306
7.0603
0.974
1.024
1.081
1.148
1.229
1.326
1.450
1.614
1.716
1.834
1.883
1.930
1.948
1.941
1.924
1.882
0.3193
0.3088
0.2877
0.2772
0.2666
0.2455
0.2350
0.1928
0.1717
0.1401
0.1190
0.0874
0.0386
0.0129
0.0073
0.0875
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
≥ 0.9999
0.9965
0.9720
0.8725
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