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Astro 1050 Mon. Apr. 3, 2017 Today: Chapter 15, Surveying the Stars

Reading in Bennett: For Monday: Ch. 15 – Surveying the Stars

Reminders: HW CH. 14, 14 due next monday.

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Chapter 12: Properties of Stars

•  How much energy do stars produce? •  How large are stars? •  How massive are stars?

– We will find a large range in properties but we need to measure distances.

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How to get distances to stars: Parallax

From Horizons by Seeds

The angular diameter here is “p” – the “parallax” in arcsec.

The linear diameter is 1 AU.

d = 206265/p in AUs

d = 1/p in units of “parsecs”

1 parsec or 1pc = 206265 AU

1 pc = 3.26 light years

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Intrinsic Brightness of Stars •  Apparent Brightness: How bright star appears to us •  Intrinsic Brightness: “Inherent” – corrected for

distance •  How does brightness change with distance?

–  Flux = energy per unit time per unit area: joule/sec/m2 = watts/m2

•  Example: 100 watt light bulb (assume this is 100 W of light energy) spread over 5 m2 desk gives 20 Watts/m2

–  Sun’s flux at the Earth •  Luminosity = 3.8 ×1026 Watts •  It has spread out over sphere of radius 1 AU = 1.5 × 1011 m

–  Surface area of sphere = 4 π R2 = 2.8 ×1023 m2 •  FSun = 3.8 ×1026 Watts / 2.8×1023 m2 = 1357 W/m2

–  Inverse Square Law: Flux falls of as 1/distance2

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Inverse-square law for light:

Inverse Square Law: Flux falls of as 1/distance2

Double distance – flux drops by 4 Triple distance – flux drops by 9

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Correcting Magnitudes for Distance •  To correct intensity or flux for distance, use Inverse Square Law

•  Up to now we have used “apparent magnitudes” mv •  Define absolute magnitude Mv as magnitude star would have

if it were at a distance of 10 pc.

•  This gives us a way to correct Magnitude for distance, or find distance if

we know absolute magnitude. Note: the book writes mv and Mv: The “V” stands for “Visual” -- Later we’ll consider magnitudes in other colors like “B=Blue” “U=Ultraviolet”

2

2

2

Bdistance

A distance

)4/()4/(

⎟⎟⎠

⎞⎜⎜⎝

⎛==

A

B

B

A

rr

rLrL

FF

ππ

pc 10 B d, distance trueA )/log(5.2 ===− ABBA IImm⎟⎟⎠

⎞⎜⎜⎝

⎛+−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛=−=−

pc 1log55

pc 10log5.2log5.2

2

d distance

pc 10pc 10

ddII

mmMm d

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=−

pc 1log55 dMm 5/)5(10 +−= Mmd

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Some Examples:

•  Fill in the Table: mV MV d (pc) P (arcsec) ___ 7 10 _______ 11 ___ 1000 _______ ___ -2 ____ 0.025 4 ___ ____ 0.040

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Some Examples:

•  Filling in the Table: m MV d (pc) P (arcsec) 7 7 10 0.1 11 1 1000 0.001 1 -2 40 0.025 4 2 25 0.040

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How to recognize patterns in data •  What patterns matter for people – and how do we recognize them? •  Weight and Height are easy to measure •  Knowing how they are related gives insight into health

•  A given weight tends to go with a given height •  Weight either very high or very low compared to trend ARE important

•  Plot weight vs. height and look for deviations from simple line

•  Example of cars from the book –  Note “main sequence” of cars –  Weight plotted backwards

•  Just make main sequence a line which goes down rather than up

–  Points off main sequence are “unusual” cars

From our text: Horizons, by Seeds

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Stars: Patterns of Lum., Temp., Rad.

•  The Hertzsprung-Russsell (H-R) diagram •  Plot L vs. Decreasing T. (We can find R given L and T)

From Horizons, by Seeds

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How are L, T, and R related? L = area ×σT4 = 4 π R2 σT4

–  Stars can be intrinsically bright because of either large R or large T

–  Use ratio equations to simplify above equation •  (Note book’s symbol for Sun is circle with dot inside)

–  Example: Assume T is different but size is same •  A star is ~ 2 × as hot as sun, expect L is 24 = 16 times as bright •  M star is ~1/2 as hot as sun, expect L is 2-4 = 1/16 as bright

•  B star is ~ 4 × as hot as sun, expect L is 44 = 256 times as bright

–  Example: Assume T same but size is different •  If a G star 4 × as large as sun, expect L would be 42=16 times as bright

42

42

42

44

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛==

SunSunSunSunSun TT

RR

TRTR

LL

σπσπ

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L, T, R, and the H-R diagram •  L = 4 π R2 σT4 •  The main sequence consists very roughly of similar size stars •  The giants, supergiants, and white dwarfs are much larger or

smaller

From our text: Horizons, by Seeds

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Lines of constant R in the H-R diagram •  Main sequence not quite

constant R •  B stars: R ~10 RSun •  M stars: R ~0.1 RSun

•  Betelgeuse: R~ 1,000 RSun

•  Larger than 1 AU

•  White dwarfs: R~ 0.01 RSun

•  A few Earth radii

•  What causes the “main sequence”? •  Why “similar” size, with

precise R related to T? •  Why range of T?

•  Why are a few stars (giants, white dwarfs) not on main sequence?

From our text: Horizons, by Seeds

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Different “types” of H-R diagrams

•  Hertzsprung-Russell diagram will appear over and over again in class

•  Deviations from patterns useful for understanding evolution of stars

•  Equivalent kinds of plots: •  Luminosity vs. Temperature (what we’ve been showing) •  Absolute Magnitude vs. Spectral Type (the “original” H-R

diagram) •  Apparent Magnitude vs. Spectral Type

•  Patterns still the same if all stars are at same difference •  All stars will be shifted vertically by the same amount:

m-M= -5 + 5 log(d) •  Magnitude vs. Color (called “color-magnitude diagrams”)

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Luminosity Classes •  Ia Bright supergiant •  Ib Supergiant •  II Bright giant •  III Giant •  IV Subgiant •  V Main sequence star

•  white dwarfs not given Roman numeral

•  Sun: G2 V •  Rigel: B8 Ia •  Betelgeuse: M2 Iab

From our text: Horizons, by Seeds

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Spectra of Different Luminosity Classes

•  Presence of different lines determined by Spectral Class (temperature) •  Width of individual lines determined by Luminosity Class

•  “Pressure broadening”: •  High density (so high pressure) ⇒frequent atomic collisions •  Energy levels shifted by 2nd nearby atom ⇒broad lines •  Main sequence stars are “high” density and pressure •  Supergiants are low density and pressure

•  Something can cause a main sequence star to expand to a large size to form a giant or supergiant

From our text: Horizons, by Seeds

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What fundamental property of a star varies along the main sequence?

•  T and R vary smoothly (and together) along the main sequence •  B stars are ~4 times hotter and ~10 times bigger than sun •  M stars are ~2 times cooler and ~10 times smaller than sun

•  Presence of a line implies that a single fundamental property is varying to make some stars B stars and some stars M stars •  That fundamental property then controls T, R •  A second property controls whether we get a giant or dwarfs

•  Fundamental properties we could measure •  Location: Doesn’t seem to be of major importance •  Composition: Outside composition of stars similar (H, He, ...) •  Age: Will be important – but put off till Chapter 9 •  Mass: Turns out to be the most important parameter

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Masses From Binary stars

Newton’s form of Kepler’s 3rd law for planets: Modified form when mass of “planet”

gets very large Dividing by same equation for Earth-Sun

and canceling constants gives:

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2 4 aGM

P π=

)(

4 32

2 aMMG

PBA +

=π

2

32 4Pa

GMM BA

π=+

2

3

yr)1/( AU) 1/(

Pa

MMM

Sun

BA =+

From our text: Horizons, by Seeds

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Masses of Binary stars

2

3

yr)1/( AU) 1/(

Pa

MMM

Sun

BA =+

From our text: Horizons, by Seeds

An example. Suppose we measured the period in a spectroscopic binary and knew the spectral types (and hence the masses, as we shall see) of the component stars. The period is 2 years (P = 2 years) and the stars are a G star (1 solar mass) and a M star (0.5 solar masses). What is the separation?

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Masses of Binary stars

2

3

yr)1/( AU) 1/(

Pa

MMM

Sun

BA =+

From our text: Horizons, by Seeds

An example. Suppose we measured the period in a spectroscopic binary and knew the spectral types (and hence the masses, as we shall see) of the component stars. The period is 2 years (P = 2 years) and the stars are a G star (1 solar mass) and a M star (0.5 solar masses). What is the separation?

MA+MB = 1.5 MSun

1.5 x (2)2 = (a/1 AU)3

6 = (a/1 AU)3

1.8 AU = a

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Measuring a and P of binaries

•  Two types of binary stars –  Visual binaries: See separate stars

•  a large, P long •  Can’t directly measure component of a along line of sight

–  Spectroscopic binaries: See Doppler shifts in spectra •  a small, P short •  Can’t directly measure component of a in plane of sky

•  If star is visual and spectroscopic binary get get full set of information and then get M

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Masses and the HR Diagram

•  Main Sequence position: – M: 0.5 MSun – G: 1 MSun – B: 40 Msun

•  Luminosity Class – Must be controlled

by something else

From our text: Horizons, by Seeds

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The Mass-Luminosity Relationship •  L = M3.5

•  Implications for lifetimes: 10 MSun star –  Has 10 × mass –  Uses it 10,000 × faster

–  Lifetime 1,000 shorter

From our text: Horizons, by Seeds

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Eclipsing Binary Stars •  System seen “edge-on” •  Stars pass in front of each other •  Brightness drops when either is

hidden

•  Used to measure: –  size of stars (relative to orbit) –  relative “surface brightness”

•  area hidden is same for both eclipses •  drop bigger when hotter star hidden

–  tells us system is edge on •  useful for spectroscopic binaries

From our text: Horizons, by Sees