Observed Properties of Stars - 2 ASTR 2110 Sarazin

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1 Observed Properties of Stars - 2 ASTR 2110 Sarazin

2 Properties Location Distance Speed Radial velocity Proper motion Luminosity, Flux Magnitudes

3 Magnitudes

4 Stellar Colors

5 Stellar Colors

6 Stellar Colors Stars vary in color: Betalgeuse red, Sun yellow, Vega blue-white Use filters to get flux in one color, compare

7 Color Filters for Observing

8 Stellar Colors Stars vary in color: Use filters to get flux in one color, compare Fluxes: F U, F B, F V, Magnitudes: m U = U, m B = B, m V = V,

9 Stellar Colors Color index, or just color CI = B V, Note: Given B V è fixed F B / F V Just measures shape of spectrum, not total flux Independent of distance Just gives color

10 Temperature & Color Color mainly determined by temperature of stellar surface Stellar spectra ~ black body λ max 0.3 cm / T (Wiens Law) Higher T è shorter λ è bluer light Hot stars blue, B V negative Cool stars red, B V positive Solar spectrum vs. BB

11 Stellar Temperatures Range from T 3000 K to 100,000 K (brown dwarfs, planets cooler; some stellar corpses hotter)

12 Stellar Temperatures

13 Bolometric Magnitude Hard to measure all light from star, but Bolometric magnitude è magnitude based on total flux m bol

14 Luminosity & Absolute Magnitude F = L 4πd 2 L = 4πd 2 F Absolute magnitude M = magnitude if star moved to d = 10 pc F = L 4πd 2 F! = 10 # F d 10 " pc $ & % 2

15 Luminosity & Absolute Magnitude F = L 4πd 2 F! = 10 # F d 10 " pc (! m M = 2.5log* 10 *# " d ) pc $ & % 2 +, $ & % = 5log ( 10 / d ) pc m M = 5logd pc 5 distance modulus

16 Luminosity & Absolute Magnitude m M = 5logd pc 5 distance modulus M = m 5logd pc + 5 = m + 5log ## π + 5

17 Luminosity & Absolute Magnitude From distance to Sun (AU) and flux M bol ( ) = L = x erg/s = x J/s = W M bol = log(l/l ) Memorize Note: = astronomical symbol for Sun

18 Stellar Luminosities Very wide range 10-4 L L 10 6 L

19 Basic Numbers of Astronomy Memorize

20 For BB, Stellar Radii L = (area) σ T 4 = 4π R 2 σ T 4 σ = 5.67 x 10-5 erg/cm 2 /s/k 4 Stefan-Boltzmann constant Define effective temperature T eff such that L = 4π R 2 σ T eff 4 If BB, then T = T eff

21 Stellar Radii Measure flux F, distance d è L Measure color èt eff (estimate) Solve for radius R

22 Stellar Radii Find mainly three sets of radii Normal Stars: main sequence, dwarfs 0.1 R < R < 20 R sequence: small, cool, faint è big, hot, bright Giants: R > 100 R ~ AU cool, T ~ 3000 K White Dwarfs: R 0.01 R ~ R(Earth)

23 Binary Stars ASTR 2110 Sarazin Albereo (β Cygni) Binary Star System

24 Types of Binary Stars Visual Binaries Spectroscopic Binaries Eclipsing Binaries

25 Visual Binaries See stars orbit one another

26 Measurements give: Measurables P = orbital period a 1, a 2, a = a 1 + a 2 semi-major axes, but angular sizes not physical sizes eccentricity e direction of LOS, inclination angle i, orientation Ω d a Measure distance d a AU = a d pc = a / π Gives orbit size in AU a"

27 Determining Masses Kepler s 3 rd M P yr 2 = a AU 3 / ( M tot / ) M [( M 1 + M 2 ) / ] = a AU 3 / P yr 2 a 1 / a 2 = a 1 / a 2 = M 2 / M 1

28 Determining Masses Example: a 1 = 1, a 2 = 4, π = 0.1, P = 100 yr, i = 0 o Masses? a = 5, a AU = a / π = 5 / 0.1 = 50 AU (M 1 + M 2 ) = a 3 AU / P 2 yr = (50) 3 / (100) 2 = 12.5 M 1 /M 2 = a 2 / a 1 = 4 / 1 = 4 M 1 = 10 primary, M 2 = 2.5 secondary M M M

29 Stellar Masses Narrower range than luminosity Roughly, 0.1 M < M < 100 M Lower masses: brown dwarfs, planets

30 Mass Luminosity Relation Main Sequence Stars: (Normal stars) (L / ) ~ (M / ) 3 Also L (R / ) ~ (M / ) 0.75 R (T eff /6000 K) ~ (M / ) 0.4 M M M Doesn t work for giants, supergiants, white dwarfs WDs Giants

31 Masses in Astronomy Mass = total amount of matter in an object Really, really want to know Mass è Gravity è motions of nearby things Method to determine mass in Astronomy Planets, Sun, moons, Stars Star clusters Galaxies Clusters of Galaxies The Universe

32

33 Visual Binaries: Types of Binary Stars Spectroscopic Binaries: See moving spectral lines from one or both stars (Doppler effect) Eclipsing Binaries:

34 Spectroscopic Binaries

35 Spectroscopic Binaries See moving spectra lines (Doppler effect) Single-lined or Double-lined Velocity Curve: Assume v << c z = Δλ / λ = v r (t)/c v r (t) = velocity curve (usually in km/s)

36 i y x Ω

37 Spectroscopic Binaries See moving spectra lines (Doppler effect) Single-lined or Double-lined Velocity Curve: Assume v << c Δλ / λ = v r (t)/c v r (t) = velocity curve (usually in km/s) v r (t) = v x (t) sin i

38 Spectroscopic Binaries v r (t) = v x (t) sin i Example: i = 90 o (edge-on), circular orbit v r (t) = v orb sin(2πt / P) sinusoidal velocity curve v orb = 2 π a / P

39 i y x Ω

40 Spectroscopic Binaries General case: i 90 o, e 0, Ω e, Ω è change shape of velocity curve, not sinusoidal v r time

41 Spectroscopic Binaries General case: i 90 o, e 0, Ω e, Ω è change shape of velocity curve, not sinusoidal Determine e, Ω from shape of velocity curve Inclination i? v r (t) = v x (t) sin i Just change all velocities by a constant factor Cannot be determined from observations!!

42 Mass Function Single-lined binary: v 1 = v orb of observed star Can determine: v r (max) = v 1 sin i, P, e, Ω (assume circular here) a 1 = v 1 P / 2π a 1 sini = (v 1 sini)p / 2π What use is this? a = a 1 + a 2 = a 1 (1+ a 2 / a 1 ) = a 1 (1+ M 1 / M 2 ) = a 1 (M 1 + M 2 ) / M 2 G(M 1 + M 2 ) 4π 2 = a3 Kepler's 3rd Law 2 P

43 Mass Function (M 1 + M 2 ) = 4π 2 a 3 = 4π 2 a 3 1 (M 1 + M 2 ) 3 GP 2 M 3 2 GP 2 M 2 3 (M 1 + M 2 ) = 4π 2 a 1 2 GP 2 (M 2 sini) 3 (M 1 + M 2 ) = 4π 2 (a 1 sini) 3 2 GP 2 = 4π 2 GP 2 3! " # v 1 (sini)p 2π = [v max 1 (obs)] 3 P 2πG $ % & 3 1 2

44 Mass Function f [v max 1 (obs)] 3 P 2πG = (M 2 sini) 3 (M 1 + M 2 ) 2 What use is this? M 2 > f (lower limit on mass of unseen star) Used to identify black holes, for example Double-lined binary: v 1 max (obs)/v 2 max (obs) = M 2 / M 1

45 Visual Binaries: Types of Binary Stars Spectroscopic Binaries: Eclipsing Binaries: Periodic changes in brightness, stars block one another stars close together, seen along orbital plane

46 Eclipsing Binaries Light Curve time

47 Eclipsing Binaries Eclipse è i 90 o edge-on 90 o i a cos i < R 1 + R 2

48 Light Curve Flux vs. time primary secondary Primary eclipse = deeper eclipse Secondary eclipse = shallower eclipse

49 Primary Eclipse Which star behind at primary eclipse? More luminous? NO! Area lost = area of smaller star = same for both eclipses Equal areas lost L = (area) σ T eff 4 change in area the same

50 Primary Eclipse Primary eclipse è hotter star behind

51 Light Curves Eccentricity: e 0 è spacing and duration of eclipses unequal Can determine e, Ω flux flux

52 Central Eclipses, Circular Orbits For simplicity, consider i = 90 o (central), e = 0 (circular) Only total or annular eclipses Constant area removedè flat bottom to eclipses

53 Central Eclipses, Circular Orbits For simplicity, consider i = 90 o (central), e = 0 (circular) Only total or annular eclipses Constant area removedè flat bottom to eclipses flux flux

54 Give radii of stars v = total orbital velocity S = smaller star L = larger star v(t 2 t 1 ) = D S = 2 R S v(t 3 t 1 ) = D L = 2 R L v = 2πa / P Eclipse Timing R S = v(t 2 t 1 )/2 = π a (t 2 t 1 ) / P R L = v(t 3 t 1 )/2 = π a (t 3 t 1 ) / P

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