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Stellar Properties
1)
Distance trig parallax d(pc) = 1/p (arcsec)
2) Velocity (Vspace)2 = (Vrad)2 + (Vtan)2
3) Brightness mag = -2.5 log (flux) + constant; L
4) Temperature B-V; spectral class
5) Mass spectroscopic binary; K, P, i
6) Radius eclipsing spectroscopic binary
tan p = 1AU / d (AU)
1. Distance
for small angles p =1 AU/ d(AU)
d (AU) = 1/p where p is in radians
1 radian = 206265 arcsec
d (AU) = 206265 / p (arcsec) (define 1pc = 206265 AU)
d (pc) = 1 / p (arcsec)
1 AU
p
d
Parallax measurements
nearest star ~ 0.8”
(d~ 1.3 pc)
ground limit ~ 0.01” (d~ 100 pc)
HST limit ~ 0.001” (d~ 1000pc)
Hipparcos (1989-1993) [120,000 stars
to 0.001”; 1 million stars to 0.02”]
GAIA (2011-2020) [1 billion stars]
2. Velocity
(Space V)2 = (Radial V)2 + (Tangential V)2
Radial V from Doppler:
/ = v/c
Tangential V from
proper motion 
arcsec/yr :
Vt = 4.74 /p km/s
proper motion
Proper Motion
Vr
sin  = = Vt/d
Vt
Vt = d = /p pc/yr
rad
d

arcsec, pc
Vt = 4.74 /p km/s
km, yr
sec


 depends on d, speed and direction
Barnard’s star (d=1.85pc) has largest =10”/yr

3. Brightness - T, size, d
Magnitude scale: backwards, logarithmic
Energy scale: luminosity, flux L=4R2T4 ergs/s
Magnitudes: each mag is factor of 2.5 fainter
1 mag = 2.5
5 mag = 100
10 mag = 10,000
mag = -2.5 log flux + constant
m2 - m1 = -2.5 log f2 / f1
Apparent mags (m) as seen from Earth
Absolute mags (M)
if object at 10 pc
m
Sun
moon
Venus
Sirius
Vega
eye
30in
5m
faintest
-26.5
-12.5
-4
-1.4
0
6
15
20
28
M
+5
+19
+27.5
+1.4
0.5
4.2
6.0
2.0
5.0
3.3
10 pc
1.3
5 pc
-26.5
2.0
5 pc
10 pc
apparent mags
15 pc
absolute mags
2.0
0.0
Absolute mag M: if star were viewed at 10pc
Apparent mag: star as viewed from earth
m-M = -2.5 log (E/d2) - (-2.5 log (E/102))
= -2.5 log E + 5 log d + 2.5 log E - 5
m-M = -5 + 5 log d distance modulus
Color Index
mb - mv = Mb - Mv = B-V
mv - mr = V-R
B-V gives temperature
Common filters: U,B,V,R,I,J,H,K Johnson
ugriz Sloan
visual
filter
Hot star looks blue B-V ~ - 0. 5
Cool star looks red B-V ~ 1. 5
T
B-V
Bolometric Magnitude:
Brightness over all  ~ L
Mbol = Mv + BC
Mbol* - Mbol = -2.5 log L*/L
Mbol ~ 4.74, L ~ 4x1033 ergs/s
QuickTime™ and a
decompressor
are needed to see this picture.
Brightness - depends on T, R, d
Magnitudes (backwards, logarithmic)= -2.5 log(flux) + C
or m2 - m1 = -2.5 log (f2/f1)
• m (apparent mag - as seen from earth - includes d)
• M (absolute mag - object at 10 pc - eliminates d)
• m-M = -5 + 5 log d (distance modulus)
• MBOL (bolometric mag - over all ) = MV + BC
• B-V (color index) - gives T
Energy (luminosity, flux)
• L= total energy from star/sec = 4R2T4 ergs/s
• Flux = energy received at earth at = L/4d2 ergs/cm2/s/Å
• MBOL* - MBOL(sun) = -2.5 log (L*/Lsun)
4. Temperature (B-V, Spectral Class)
Mv
-6
+1
+5
+15
Class Lines
O(5-9)
B(0-9)
A
F
G
K
M
R
N
S
L
T
Temp
>30,000K
HeII
11-30,000
He
H strong 7500-11000
6000-7500
CaII
metals 5000-6000
metals,bands
3500-5000
TiO
2000-3500
B-V
B-V=-0.865 + 8540/T
-0.3
-0.1
0.0
0.3
0.6
1.0
1.5
K with C
M with C
ZrO
hydrides <2000
methane <1300
sun = G2V
Luminosity Class: I, II=SG, III, IV=Giant, V=dwarf (main sequence)
Spectral Class Mnemonics
Oh, Be A Fine Girl(Guy), Kiss Me Right Now Smack
Oh Brother, Astronomy Finally Gruesomely Killed Me
Right Now *Slump*
Oven Baked Ants, Fried Gently, Kept Moist,
Retain Natural Succulence (Largely True)
He
vega
H
sun
metals
Betelgeuse
molecules
Info from Spectra:
• abs= normal star, emission = disk or jet
•
composition of outer layers (if line present, element present
•
temperature of outer layers (from knowledge of energy levels of element)
• density (narrow lines imply low density)
• pressure (wide lines imply high pressure)
• rotation (high rotation makes wider lines)
• binarity (see spectra of two different stars)
• wind (strange P Cygni line profiles with absorption + emission)
• magnetic field (Zeeman splitting of lines)
WD spectrum
Spectra of giants
P Cygni features
Spectroscopic parallax:
1. Use stars < 100pc to calibrate MV for spectral classes
2. For unknown star:
a) use CCD to measure mV
b) use spectrograph to fine spectral class
c) use calibration from (1) to get MV
d) use distance modulus to calculate d
5. Mass (double - lined spectroscopic binaries)
m1/m2 = v2/v1
m1 + m2 = 42a3/GP2 (a=vP/2)
v1 sin i, v2 sin i, P
come from radial velocity curve of binary
Alcor and Mizar are just neighbors but Mizar itself is a
visual binary and Mizar A and Mizar B are each binaries
d1
m1
d2
m2
x
center of mass physics
m1d1 = m2d2
v = 2d/P so d=vP/2
m1v1P/2 = m2v2P/2
m1v1 = m2v2
m1/m2 = v2/v1
Kepler’s 3rd law
Fg = Fc
GmM/r2 = mv2/ r
GM/r = 42r2/P2
M = 42r3/P2
v = 2r/P
m
.
r
.
M
Mass - from spectroscopic binaries need K1, K2, P, i)
m1/m2 = v2 / v1= K2 / K1
m1 + m2 = 42(a1 + a2)3 /GP2
for double-lined binary
K1 = v1 sin i = 2a1sin i / P
a1 = PK1 / 2 sin i
a1 + a2 = P (K1 + K2) / 2 sin i
m1 + m2 = (42 / GP2)(K1+K2)3/83sin3i = P(K1+K2)3/2G sin3i
(m1+ m2)P2 = (a1+ a2)3 = a13(1+ a2/a1)3
For single-lined binary
with solar mass units
a2/a1 = m1/m2
(m1+ m2)P2 = a13(1+ m1/m2)3 = a13(m1+ m2)3/m23
f(m1, m2) = m23sin3i/(m1+m2)2 = a13/P2 = K13P/83
mass function gives a lower limit to m2
Mass of Sun (from planet orbits) = 2 x 1033 g
Star masses range from 0. 07 M to 100 M
Which is star 1 and which star 2?
Which star is more massive?
K1
K2
m2/m1 = K1/K2 ~ ?
6. Radius
•
from lunar occultation
•
from interferometry (for supergiants)
•
from T, L (R = [L/4T4]1/2)
•
from eclipsing, spectroscopic binaries (need eclipse
times, K1, K2)
D1 = (K1+ K2) ta-b where a-b is ingress or egress time
D2 = (K1+ K2) ta-c where a-c is ingress/egress + eclipse time
Radii of stars range from 1/100 R to 400 R
a
a
b
d
c
primary eclipse
b
d
secondary eclipse
c
a-b or c-d moves diameter of small star Ds
a-c or b-d moves diameter of large star DL
Ds = V x ta-b
DL = V x ta-c where V = Ks + KL
P=?
e=?
i=?
Rs/RL= ?
Ms/ML=?
a =?
Ds=?
DL=?
Ms=?
ML=?
Ls/LL=?
1
5
10
15
20
25
Ts/TL=?