stellar properties distance trig parallax d(pc) = 1/p (arcsec)

Stellar Properties

1) Distance trig parallax d(pc) = 1/p (arcsec)

ü Velocity (Vspace)2 = (Vrad)2 + (Vtan)2

1) Brightness mag = -2.5 log (flux) + constant; L

2) Temperature B-V; spectral class

3) Mass spectroscopic binary; K, P, i

4) Radius eclipsing spectroscopic binary

pd

1 AU

tan p = 1AU / d (AU)

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. Distance

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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 (2013-2018) [1 billion stars] to 0.000020”

2. Velocity

proper motion

(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

d

Vt

Vr

sin = = Vt/d

Vt = d = /p pc/yr

rad arcsec, pc km, yr sec

Vt = 4.74 /p km/s

depends on d, speed and direction

Barnard’s star (d=1.85pc) has largest =10”/yr

http://youtu.be/yxPPDDP5kyQ




3. Brightness - T, size, dMagnitude scale: backwards, logarithmic

Energy scale: luminosity, flux L=4R2T4 ergs/s

Magnitudes: each mag is factor of 2.5 fainter

1 mag = 2.5

5 mag = 100

10 mag = 10,000

Apparent mags (m) as seen from Earth

Absolute mags (M) if object at 10 pc

mag = -2.5 log flux + constant

m2 - m1 = -2.5 log f2 / f1

Sun -26.5 +5

moon -12.5 +19

Venus -4 +27.5

Sirius -1.4 +1.4

Vega 0 0.5

eye 6

30in 15

5m 20

faintest 28

m M

apparent mags absolute mags

5 pc

10 pc

15 pc

5 pc

10 pc

-26.5

1.3

2.0

3.3

4.2

5.06.0

2.0 2.00.0

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

Hot star looks blue B-V ~ - 0. 5

Cool star looks red B-V ~ 1. 5

visual filter

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

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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 = 4R2T4 ergs/s

• Flux = energy received at earth at = L/4d2 ergs/cm2/s/Å

• MBOL* - MBOL(sun) = -2.5 log (L*/Lsun)

4. Temperature (B-V, Spectral Class)

-6 O(5-9) HeII >30,000K -0.3

B(0-9) He 11-30,000 -0.1

+1 A H strong 7500-11000 0.0

F CaII 6000-7500 0.3

+5 G metals 5000-6000 0.6

K metals,bands 3500-5000 1.0

+15 M TiO 2000-3500 1.5

R K with C

N M with C

S ZrO

L hydrides <2000

T methane <1300

Mv Class Lines Temp B-V

Luminosity Class: I, II=SG, III, IV=Giant, V=dwarf (main sequence)

sun = G2V

B-V=-0.865 + 8540/T

T~ 9000/[(B-V)+0.93]

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

H

metals

molecules

Sun

Vega

Betelgeuse

Jacoby atlas

1984, ApJS,

56, 257

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 find spectral class

c) use calibration from (1) to get MV

d) use distance modulus to calculate d

Different Kinds of Temperature

Type From Observe

Brightness Planck fctn F

Color Planck fctn B-V

Effective (T4) Stefan-Boltzman L & R

Excitation Boltzman Ratio of lines

Ionization Saha Ratio of lines

Kinetic Thermal Doppler Width of lines

5. Mass (double - lined spectroscopic binaries)

m1/m2 = v2/v1

m1 + m2 = 42a3/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

xm1m2

d1 d2 m1d1 = m2d2

v = 2d/P so d=vP/2

m1v1P/2 = m2v2P/2

m1v1 = m2v2

m1/m2 = v2/v1

center of mass physics

Kepler’s 3rd law

Fg = Fc

GmM/r2 = mv2/ r v = 2r/P

GM/r = 42r2/P2

M = 42r3/GP2

.r. Mm

Mass - from spectroscopic binaries need K1, K2, P, i)

m1/m2 = v2 / v1= K2 / K1

m1 + m2 = 42(a1 + a2)3 /GP2

K1 = v1 sin i = 2a1sin i / P

a1 = PK1 / 2 sin i

a1 + a2 = P (K1 + K2) / 2 sin i

m1 + m2 = (42 / GP2)P3(K1+K2)3/83sin3i = P(K1+K2)3/2G sin3i

for double-lined binary

(m1+ m2)P2 = (a1+ a2)3 = a13(1+ a2/a1)3

a2/a1 = m1/m2

(m1+ m2)P2 = a13(1+ m1/m2)3 = a1

3(m1+ m2)3/m23

f(m1, m2) = m23sin3i/(m1+m2)2 = a1

3/P2 = K13P/83

For single-lined binary with solar mass units

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?

K2K1

m2/m1 = K1/K2 ~ ?

6. Radius

• from lunar occultation

• from interferometry (for supergiants)

• from T, L (R = [L/4T4]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

b c

d

a b cd

primary eclipse secondary eclipse

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

1 5 10 15 20 25

P=?

e=?

i=?

Rs/RL= ?

Ms/ML=?

a =?

Ds=?

DL=?

Ms=?

ML=?

Ls/LL=?

Ts/TL=?

1 5 10 15 20 25

P=24 hr

e=0

i=90

Rs/RL= 1/3

Ms/ML=2

a= 3x106=.02AU

Ds=8.1x105=0.6Dsun

DL=2.4x106=1.7Dsun

Ms=0.4Msun

ML=0.8Msun

Ls/LL=1.51

Ts/TL=1.9

radius

density

0.01R

400 R

10-6 g/cm3

106 g/cm3

mass

100 M

0.07M

Location depend on:

Mass

Age

Composition

uses ~20,000 stars

Mass - Luminosity Relation

stellar properties distance trig parallax d(pc) = 1/p (arcsec)

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