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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
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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:
32
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
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