Physics 556 Stellar Astrophysics Prof. James Buckley Lecture 1 Mapping the Universe

Physics 556 Course Outline • Stellar Structure and Evolution (and other topics in Astrophysics) • Crow 206, 11:30-1:00 • Instructor: Professor James Buckley (no TA) • Office: Compton 253 • Office hours: TBD • Textbook: No official textbook - course notes. Useful books include: • Radiative Processes in Astrophysics, Rybicki and Lightman,

Principles of Stellar Evolution and Nucleosynthesis, Clayton, Quarks and Leptons, Halzen and Martin, Advanced Stellar Astrophysics, William K.

• Course requires a knowledge of undergraduate E&M, Quantum

Mechanics, Mechanics and Statistical Physics (but will review material as it arrises)

• Grade based on one midterm (30%), homework (40%) and final exam project (30%). Class attendance is required.

Physics 556 Syllabus • Introduction:

historical background, astronomical coordinates, distance, stellar

magnitudes.

• Theory of radiation. • Statistical physics and thermodynamics. • Stellar Structure: hydrostatic equilibrium, radiative transfer, convective transfer, nuclear burning, the Lane-Emden equation for Polytropes.

• Relativistic quantum mechanics, Dirac equation, fermions and bosons, quantum statistics, Einstein coefficients.

• Reaction equilibrium, ionization equilibrium, out-of-equilibrium processes • Equations of state, degenerate matter. • Time dependent perturbation theory, electromagnetic and weak interactions. • Stellar opacity and radiation absorption processes, oscillator strength, bound-bound, bound-free, free-free interactions

• Weak interactions, neutrinos, beta-decay. • Nuclear fusion, interaction rates, WKB approximation. • Stellar stability and evolution. • Accretion power, supernovae and pulsar magnetospheres (as time permits) • White dwarfs, neutron stars and black holes

Reading

($113-$145 new Amazon)

($24-$45 on Amazon)

($40-$80 used on Amazon)

($27-$60 used on Amazon)

• I recommend purchasing Clayton’s book since it is not very expensive and probably is the single best reference. Radiative Processes is an important one to have on your shelf, but pretty pricy. I put this on reserve - you might be able to copy the relevant chapters (mostly chapter 1). Rose is a reasonable reference but not a very good textbook - you can pick it up used for not too much. Halzen and Martin is an important book for many classes, but you can probably get away with class notes and checking the reserve book.

• Diurnal motion: stars and sun rise in east and set in the

west.

Diurnal Motion of Stars

Polaris

West

North

East

• To a terrestrial observer stars seem affixed to a celestial sphere that revolves around an axis pointing in the direction of Polaris

Physics 312 - Lecture 1

– p.3

rizon Coordinates Horizon Coordinates 3,-%4(!"#"$%&'#(5,#" 567 8.0 • 9&: *+ 23/.2;

sin a sin b sin c = = sin A sin B sin C

Angular Distance

Angular Distance • !"#$ A #" (α, δ)% !"#$ B #" (α + ∆α, δ + ∆δ) !"

&

1 !#

%

!$

2 #

V '()(*"+#),-./#"0$

"

sin(∆α) sin φ = sin(∆θ) sin [90◦ − (δ + ∆δ)] sin(∆α) cos(δ + ∆δ) = sin(∆θ) sin φ sin φ ∆α ≈ ∆θ cos δ

Angular Distance

Angular Distance

ance

!"#$%#&%#' $" &() $*) (+,-- ,#'-) ,../"0%+,$%"#1 "#) 2,# 3/%$) $ B #" (α + ∆α, δ,#+)0./)((%"# ∆δ) 4"/ $*) 2*,#') %# 5)2-%#,$%"# ,#5 2"+6%#) $*) /)(&-$(7 !"

&

1 !#

%

2 #

V '()(*"+#),-./#"0$

"

!$

∆δ = ∆θ cos φ ∆θ sin φ = ∆α cos δ (∆θ)2 cos2 φ + (∆θ)2 sin2 φ = (∆α cos δ)2 + (∆δ)2

8),5%#' $" $*) %+."/$,#$ /)(&-$ $*,$ $*) ,#'&-,/ 5%($,#2) ∆θ 6)$3))# $3" ."%#$( sin φ 5%44)/%#' %# 9: ,#5 ;

n(∆α) = n(∆θ) sin [90◦ − (δ + ∆δ)] + ∆δ) = sin(∆θ) sin φ (∆θ)2 ≈ (∆α cos δ)2 + (∆δ)2 sin φ ∆α ≈ ∆θ

Galactic Coordinates http://en.wikipedia.org/wiki/File:Galactic_coordinates.JPG

Aristarchus of Samos • Aristarchus lived on the Greek island of Samos from 310 BC to 230 BC.

• First to postulate that the planets orbited the Sun - not the Earth

• Estimated size of the Earth, size and distance to our Moon, the size and distance to our Sun

• Deduced that the points of light we see at night are not dots painted on some celestial sphere but stars like our Sun at enormous distances.

”Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface. “ - Archimedes

Diameter of the Earth • Erastothenes of Cyrene (modern day Libya) (276-194 BC) was a Greek mathematician, astronomer, Librarian of Alexandria, friend of Archimedes.

• Sun visible at bottom of a well, vertical sticks cast no shadow

in Syene on the summer solstice at local noon. In Alexandria, on the same day, a a stick cast a measurable shadow.

7.2 deg

Alexandria

• From measurements of shadows in Alexandria, the angle of

elevation of the Sun corresponded to 1/50 of a full circle (7°12') south of the zenith at the same time.

• Assuming Alexandria was due north of Syene, distance from

Alexandria to Syene must be 1/50 of circumference of the Earth. The estimated distance between the cities was 5000 stadia. He rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion is uncertain, but was likely about 185 m, which implies a circumference of 46620 km, only 16.3% too large.

7.2 deg

Syen

Distance to Sun

3°

87°

• Aristarchus argued, if one measured the angle between the moon and sun when the moon is

exactly half illuminated then one could compute the ratio of their distances. Aristarchus estimated the angle at half illumination ≈ 87° so the ratio of the distances is sin(3°). (Note: Degrees and trigonometry had not been invented yet) Aristarchus used an approximation (for what we call sin) and obtained the inequality: 1/18 > sin 3 > 1/20

• He deduced that the sun was between 18 to 20 times as far away as the moon. In fact at the

moment of half illumination the angle between the moon and the sun is actually 89 50' and the sun is actually about 400 times further away than the moon. Knowing the ratio of distances, and the relative angular sizes Aristarchus also deduced the radius of the Sun and Moon.

Hipparchus (190-120 BC)

• Hipparchus measured the distance from the earth to the moon • During a total eclipse in Syene, an observer in Alexandria saw 1/5 of the sun • Angular size of the moon and sun is ~0.5°, so θ=1/10° • Knowing the distance D between the two cities, he derived the distance to the moon from D=Dm θ

The Dark Ages

Aristotle, 384-322BC

Ptolemy, 85-165AD

The Church

•

Aristotle said “If the stars affixed to the celestial sphere are not centered on the earth, and the earth is rotating around the sun, we should see some displacement of the stars. We don’t see this, so unless the stars are ridiculously far away, the earth is the center of the universe!”

•

Aristotle also said “The natural state of a body is to be at rest, and only the presence of a force or impulse would move it. Therefore a heavy body should fall faster than a light one, because it would have a greater pull towards Earth.”

•

Aristotle and Ptolemy prevailed with their Geocentric model of the universe until the 16th century. Everything was OK except for those darned “wanderers” or planets.

History

The Copernican model Nicolaus Copernicus discovered that the seemingly complicated motions of the planets could be explained if the sun was stationary and the planets Mercury, Venus, Earth, Mars, Jupiter and Saturn orbited about the sun.

400th Anniversary of Galileo

Galileo

Used new invention, telescope, to observe jupiter

If moons orbit jupiter, why not earth orbit the sun?

History

Newton’s laws

• In 1687 Newton published Philosophiae Naturalis Principia

Mathematica • A body at rest stays at rest, a body in motion stays in motion with

the same velocity unless acted upon by an external force.

Distance to Planets Parallax distance S un E arth M ercu ry !

Venus transit, APOD July 20, 2004

Relative scales of the solar system Planet

Period (years)

Approx. Radius (a.u.)

Earth

1.0

1.0

Mercury

0.241

0.39

Venus

0.615

0.72

Mars

1.881

1.5

Jupiter

11.86

5.2

Physics 312 - Lecture 3

– p.9/12

Venus transit, Crow Observatory, June 5, 2012

Summary

•

Eclipses imply earth is round, sun and moon have about the same angular size

•

Length of shadows at different latitudes was used to measure radius of the earth

•

Distance to Venus and Mercury were originally determined by parallax during transits using the known radius of the earth as the baseline

•

Radar ranging of the planets (Mars) is now more precise

•

Knowing the distance to one planet determines value of 1 AU, and scale of entire solar system: 1 AU = 1.496×108 km = 93 million miles

•

Can determine mass of the sun to be 1.989×1030 kg from Newton’s law

GM! m mv 2 = r r2

v = 2πr/P

Parallax Distance 1 AU 1 AU d= ≈ tan p p

1 pc d≈ p”

• By definition, a star at a distance of one parsec (1 pc) will have a parallax angle of one arcsec (1”)

• 1 pc = 3.08568025 x 1018 cm • 1 pc = 3.26163626 ly • Nearest star, Proxima Centauri,

has a parallax angle of 0.77” and a distance of 1.3 pc or 4.2 ly

5 References 6 External links

Nearest Stars

List Designation

#

1

System

Star

Solar System

Sun

Alpha Centauri (Rigil Kentaurus; Toliman)

Declination [2]

variable: the Sun travels along the ecliptic

Distance[4] Light-years (±err)

Additional references

180°

0.000015

has 8 planets

11.09[2] 15.53[2]

14h 29m 43.0s

!62° 40! 46" 0.768 87(0 29)"[5][6] 4.2421(16)

# Centauri A (HD 128620)

2

G2V [2]

0.01[2]

14h 39m 36.5s

!60° 50! 02"

# Centauri B (HD 128621)

2

K1V [2]

1.34[2]

5.71[2]

14h 39m 35.1s

!60° 50! 14"

4

M4.0Ve

9.53[2]

13.22[2]

17h 57m 48.5s

+04° 41! 36" 0.546 98(1 00)"[5][6] 5.9630(109)

5 M6.0V[2] 13.44[2] 16.55[2]

10h 56m 29.2s

+07° 00! 53" 0.419 10(2 10)"[5]

6 M2.0V[2]

11h 03m 20.2s

+35° 58! 12" 0.393 42(0 70)"[5][6] 8.2905(148)

06h 45m 08.9s

!16° 42! 58" 0.380 02(1 28)"[5][6] 8.5828(289)

01h 39m 01.3s

!17° 57! 01" 0.373 70(2 70)"[5]

Wolf 359 (CN Leonis)

4

Lalande 21185 (BD+36°2147)

Luyten 726-8

Right ascension[2]

Parallax[2][3] Arcseconds(±err)

M5.5Ve

3

6

G2V [2] !26.74 [2] 4.85[2]

Epoch J2000.0

1

Barnard's Star (BD+04°3561a)

5

Apparent Absolute magnitude magnitude (m V ) (M V )

Proxima Centauri (V645 Centauri)

2

Sirius (# Canis Majoris)

Star #

Stellar class

4.38[2]

0.747 23(1 17)"[5][8] 4.3650(68)

7.47[2]

10.44[2]

7

A1V [2]

!1.46 [2]

1.42[2]

Sirius B

7

DA2 [2]

8.44[2]

11.34[2]

Luyten 726-8 A (BL Ceti)

9

M5.5Ve

12.54[2] 15.40[2]

Luyten 726-8 B (UV Ceti)

10

M6.0Ve

12.99[2] 15.85[2]

Sirius A

7.7825(390)

8.7280(631)

7

Ross 154 (V1216 Sagittarii)

11

M3.5Ve

10.43[2] 13.07[2]

18h 49m 49.4s

!23° 50! 10" 0.336 90(1 78)"[5][6] 9.6813(512)

8

Ross 248 (HH Andromedae)

12

M5.5Ve

12.29[2] 14.79[2]

23h 41m 54.7s

+44° 10! 30" 0.316 00(1 10)"[5]

3.73[2]

6.19[2]

03h

7.34[2]

9.75[2]

23h 05m 52.0s

!35° 51! 11" 0.303 64(0 87)"[5][6] 10.742(31)

15 M4.0Vn

11.13[2] 13.51[2]

11h 47m 44.4s

+00° 48! 16" 0.298 72(1 35)"[5][6] 10.919(49)

16

13.33[2] 15.64[2]

9

Epsilon Eridani (BD!09°697)

13

K2V [2]

10

Lacaille 9352 (CD!36°15693)

14

M1.5Ve

11

Ross 128 (FI Virginis)

12

EZ Aquarii (GJ 866, Luyten 789-6)

EZ Aquarii A

M5.0Ve

32m

55.8s

!09° 27! 30" 0.309 99(0

79)"[5][6]

[2] 15.58[2] M? !15°encyclopedia 18! 07" 0.289 50(4 40)"[5] 13.27 22h 38m the 33.4s free List 16 of nearest stars - Wikipedia,

EZ Aquarii B EZ Aquarii C

[7]

16

M?

14.03[2] 16.34[2]

10.322(36) has two 10.522(27) proposed planets

11.266(171)

Magnitude Scale M=1

M=2

M=3

M=4

M=5 M=6

• Hipparchus (followed by Ptolemy) created a catalog of Brighter Star (to be measured) Reference Star

about 1000 stars that were grouped into six magnitude groups. Ptolemy called the brightest stars first magnitude or m=1, the second brightest m=2 and so on.

• In the early 19th century, William Herschel (born in Aperture D1 Aperture D2

Hanover, Germany 1738 - built massive 48” reflector and 20’ refractor) devised a naked-eye method to make quantitative measurements of magnitude

• Herschel’s method consisted of viewing a reference star (with a stopped-down telescope) and an unknown with a star (with an identical telescope). When the aperture was adjusted so that the apparent magnitudes were the same, the apparent magnitude could be determined:

F1 · πD12 /4 = F2 · πD22 /4

F1 /F2 = (D2 /D1 )2

Stellar Magnitudes • In 1856, Pogson made more precise measurements verifying Hershell’s result that a first magnitude star is about 100 times brighter than a 6th magnitude star.

• Pogson formalized the system, the ratio of brightness of two stars with apparent

magnitude differing by 1, was defined to be exactly 1001/5=2.512, now known as the Pogson ratio.

• Pogson’s scale was originally fixed by assigning Polaris a magnitude of 2.

When Polaris

was found to be variable, Vega became the standard reference with m=0.

• Some examples:

The sun has m=-26.73, the full moon m=-12.6, maximum brightness of Jupiter m=-2.94, brightest star Sirius m=-1.47, Vega m=0.03, Andromeda galaxy m=3.44

• The absolute magnitude M of a star is defined so that it is the same as the apparent

magnitude for stars at a distance of 10 pc (typical for nearby stars). From your textbook: 1001/5 = 100.4 = 2.512

F1 /F2 = 100.4(m2 −m1 ) Fabs /Frel = r2 /(10 pc)2 = 100.4(m−M ) M = m + 5 − 5 log r

Solar Luminosity

L⊙ = 3.836 × 1033 erg s−1

L⊙ 6 −1 −2 Radient flux at Earth : F = = 1.36 × 10 erg s cm 4πd2

Solar constant : F = 1.36 kW m−2

Standard Candles • Understanding stellar structure is a key to finding “standard candles” • Even if we can’t determine the absolute luminosity of a particular type of star, can use as a standard candle if we:

✴ know the luminosity is a constant (not usually the case) ✴ know the dependence of luminosity on some parameter (e.g., color, spectral lines, period of oscillation, etc.) ✴ have a calibration through a parallax distance to some representative members of this class

• One motivation of understanding stellar structure, is to be able to use stars to map out the large scale structure and evolution of the universe.

• A classic example of a standard candle is a “Cepheid variable”.

For these stars the luminosity varies periodically, with the absolute luminosity tightly correlated with the period. Getting parallax distances for a few of these, calibrates the use of Cepheids as standard candles for distance measurements.

Tully Fisher Relation

• In 1977 astronomers Brent Tully and Richard Fisher determined an

empirical relationship between intrinsic luminosity and rotation velocity of spiral galaxies.

• Rotation velocities are readily measured by Doppler shifts of spectral lines.

• As for all standard candles, must calibrate some representative objects by another distance measure (e.g., Cephied variables) and can then use the apparent brightness and inverse square law to determine distance

Distances in our Galaxy • Transitions in relative spins of electrons and protons in

neutral hydrogen give rise to 21 cm (1420 MHz) radiation

p

e

➱

p

e

• Molecular clouds containing hydrogen, CO, etc. rotate around GC in Keplerian orbits.

• Doppler shifted 21 cm line gives line-of-sight velocity -

can use Kepler’s laws to reconstruct distribution of matter in galaxy, distances, enclosed mass (Dark Matter!)

Sun

• Association of galactic objects (e.g., supernova remnants, pulsars) with molecular clouds can give a crude distance (sometimes the only distance) to galactic objects.

Distance http://abyss.uoregon.edu/~js/ast123/lectures/lec13.html

• Parallax to kpc • Spectroscopic Parallax • Cepheids to 20 Mpc • Tully-Fisher relation to > 100 Mpc • Type Ia supernovae to > 1000 Mpc

Standard Candles and Redshift • If we know the intrinsic luminosity of stars, and measure their brightness we can measure distance

• Often hard to know the intrinsic luminosity.

Objects for which we have some basis of calibrating intrinsic luminosity are known as standard candles and can be used for measuring distance (or rigorously luminosity distance)

• Hubble observed more distant (fainter) galaxies appeared to recede more quickly. Now standard candles can be used to calibrate the redshift-distance relationship (Hubble’s law) to map the most distant universe.

Edwin Hubble, born 1989 in Marshfield, MO!

Redshift /Velocity Relationship A modern Hubble diagram

λobs = λrest

B A R(t)

�

1 + vr /c 1 − vr /c

λobs − λrest Redshift : z ≡ λrest � 1 + vr /c z= −1 1 − vr /c 1 + vr /c 1 + 2vr /c + (vr /c)2 v/c � 1 ⇒ = ≈ 1 + 2vr /c 2 1 − vr /c 1 − (vr /c) [1 + 2vr /c]

1/2

vr z≈ c

H0 d vr = c c

1 ≈ 1 + 2vr /c = 1 + vr /c 2

What are Stars made of?

Spectra

• Spectroscopy is the key to understanding the composition of stars, stellar structure, physical parameters of stars

Spectral Classification

Annie Canon observing in 1895. At the time Wellesley students observed using a 4-inch Browning telescope that could be set up on the north or south porch of College Hall.

•

Harvard group expanded upon the spectral classification subdividing into 15 types labeled A through O according to the strength of the Balmer lines.

•

M.N. Saha, an Indian physicist showed how, for a given temperature, one could calculate the most likely energy level at which an atom's electrons could be found.

•

Cecilia Payne used Saha's work to show how the strength of hydrogen lines depends on temperature.

•

Annie Canon, who actually led the Harvard classification project, reorganized the classification scheme based on this work, rearranging the classes from hot to cold – O, B, A, F, G, K, M.

Spectral Classification

Spectral Classification

that relate the two systems:

CGS Units

1 Ba = 1 g/(cm·s2 ) = 10-3 kg/(10-2 m·s2 ) = 10-1 kg/(m·s 2 ) = 10-1 Pa.

Definitions and conversion factors of CGS units in mechanics CGS unit

CGS unit abbreviation

Symbol

length, position

L, x

mass

m

gram

g

1/1000 of kilogram = 10!3 kg

time

t

second

s

1 second

=1s

velocity

v

centimetre per second

cm/s

cm/s

= 10!2 m/s

force

F

dyne

dyn

g cm / s 2

= 10!5 N

energy

E

erg

erg

g cm2 / s 2

= 10!7 J

power

P

erg per second

erg/s

g cm2 / s 3

= 10!7 W

pressure

p

barye

Ba

g / (cm s 2 )

= 10!1 Pa

dynamic viscosity

μ

poise

P

g / (cm s)

= 10!1 Pa·s

wavenumber

k

kayser

cm!1

cm!1

= 100 m !1

centimetre

cm

Definition

Equivalent in SI units

Quantity

1/100 of metre

= 10!2 m

Centimeter gram second (cgs) system of units - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Centimetre_gram_second_system_of_units

Pag

Physics 556 Course Outline • Stellar Structure and Evolution (and other topics in Astrophysics) • Crow 206, 11:30-1:00 • Instructor: Professor James Buckley (no TA) • Office: Compton 253 • Office hours: TBD • Textbook: No official textbook - course notes. Useful books include: • Radiative Processes in Astrophysics, Rybicki and Lightman,

Principles of Stellar Evolution and Nucleosynthesis, Clayton, Quarks and Leptons, Halzen and Martin, Advanced Stellar Astrophysics, William K.

• Course requires a knowledge of undergraduate E&M, Quantum

Mechanics, Mechanics and Statistical Physics (but will review material as it arrises)

• Grade based on one midterm (30%), homework (40%) and final exam project (30%). Class attendance is required.

Physics 556 Syllabus • Introduction:

historical background, astronomical coordinates, distance, stellar

magnitudes.

• Theory of radiation. • Statistical physics and thermodynamics. • Stellar Structure: hydrostatic equilibrium, radiative transfer, convective transfer, nuclear burning, the Lane-Emden equation for Polytropes.

• Relativistic quantum mechanics, Dirac equation, fermions and bosons, quantum statistics, Einstein coefficients.

• Reaction equilibrium, ionization equilibrium, out-of-equilibrium processes • Equations of state, degenerate matter. • Time dependent perturbation theory, electromagnetic and weak interactions. • Stellar opacity and radiation absorption processes, oscillator strength, bound-bound, bound-free, free-free interactions

• Weak interactions, neutrinos, beta-decay. • Nuclear fusion, interaction rates, WKB approximation. • Stellar stability and evolution. • Accretion power, supernovae and pulsar magnetospheres (as time permits) • White dwarfs, neutron stars and black holes

Reading

($113-$145 new Amazon)

($24-$45 on Amazon)

($40-$80 used on Amazon)

($27-$60 used on Amazon)

• I recommend purchasing Clayton’s book since it is not very expensive and probably is the single best reference. Radiative Processes is an important one to have on your shelf, but pretty pricy. I put this on reserve - you might be able to copy the relevant chapters (mostly chapter 1). Rose is a reasonable reference but not a very good textbook - you can pick it up used for not too much. Halzen and Martin is an important book for many classes, but you can probably get away with class notes and checking the reserve book.

• Diurnal motion: stars and sun rise in east and set in the

west.

Diurnal Motion of Stars

Polaris

West

North

East

• To a terrestrial observer stars seem affixed to a celestial sphere that revolves around an axis pointing in the direction of Polaris

Physics 312 - Lecture 1

– p.3

rizon Coordinates Horizon Coordinates 3,-%4(!"#"$%&'#(5,#" 567 8.0 • 9&: *+ 23/.2;

sin a sin b sin c = = sin A sin B sin C

Angular Distance

Angular Distance • !"#$ A #" (α, δ)% !"#$ B #" (α + ∆α, δ + ∆δ) !"

&

1 !#

%

!$

2 #

V '()(*"+#),-./#"0$

"

sin(∆α) sin φ = sin(∆θ) sin [90◦ − (δ + ∆δ)] sin(∆α) cos(δ + ∆δ) = sin(∆θ) sin φ sin φ ∆α ≈ ∆θ cos δ

Angular Distance

Angular Distance

ance

!"#$%#&%#' $" &() $*) (+,-- ,#'-) ,../"0%+,$%"#1 "#) 2,# 3/%$) $ B #" (α + ∆α, δ,#+)0./)((%"# ∆δ) 4"/ $*) 2*,#') %# 5)2-%#,$%"# ,#5 2"+6%#) $*) /)(&-$(7 !"

&

1 !#

%

2 #

V '()(*"+#),-./#"0$

"

!$

∆δ = ∆θ cos φ ∆θ sin φ = ∆α cos δ (∆θ)2 cos2 φ + (∆θ)2 sin2 φ = (∆α cos δ)2 + (∆δ)2

8),5%#' $" $*) %+."/$,#$ /)(&-$ $*,$ $*) ,#'&-,/ 5%($,#2) ∆θ 6)$3))# $3" ."%#$( sin φ 5%44)/%#' %# 9: ,#5 ;

n(∆α) = n(∆θ) sin [90◦ − (δ + ∆δ)] + ∆δ) = sin(∆θ) sin φ (∆θ)2 ≈ (∆α cos δ)2 + (∆δ)2 sin φ ∆α ≈ ∆θ

Galactic Coordinates http://en.wikipedia.org/wiki/File:Galactic_coordinates.JPG

Aristarchus of Samos • Aristarchus lived on the Greek island of Samos from 310 BC to 230 BC.

• First to postulate that the planets orbited the Sun - not the Earth

• Estimated size of the Earth, size and distance to our Moon, the size and distance to our Sun

• Deduced that the points of light we see at night are not dots painted on some celestial sphere but stars like our Sun at enormous distances.

”Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface. “ - Archimedes

Diameter of the Earth • Erastothenes of Cyrene (modern day Libya) (276-194 BC) was a Greek mathematician, astronomer, Librarian of Alexandria, friend of Archimedes.

• Sun visible at bottom of a well, vertical sticks cast no shadow

in Syene on the summer solstice at local noon. In Alexandria, on the same day, a a stick cast a measurable shadow.

7.2 deg

Alexandria

• From measurements of shadows in Alexandria, the angle of

elevation of the Sun corresponded to 1/50 of a full circle (7°12') south of the zenith at the same time.

• Assuming Alexandria was due north of Syene, distance from

Alexandria to Syene must be 1/50 of circumference of the Earth. The estimated distance between the cities was 5000 stadia. He rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion is uncertain, but was likely about 185 m, which implies a circumference of 46620 km, only 16.3% too large.

7.2 deg

Syen

Distance to Sun

3°

87°

• Aristarchus argued, if one measured the angle between the moon and sun when the moon is

exactly half illuminated then one could compute the ratio of their distances. Aristarchus estimated the angle at half illumination ≈ 87° so the ratio of the distances is sin(3°). (Note: Degrees and trigonometry had not been invented yet) Aristarchus used an approximation (for what we call sin) and obtained the inequality: 1/18 > sin 3 > 1/20

• He deduced that the sun was between 18 to 20 times as far away as the moon. In fact at the

moment of half illumination the angle between the moon and the sun is actually 89 50' and the sun is actually about 400 times further away than the moon. Knowing the ratio of distances, and the relative angular sizes Aristarchus also deduced the radius of the Sun and Moon.

Hipparchus (190-120 BC)

• Hipparchus measured the distance from the earth to the moon • During a total eclipse in Syene, an observer in Alexandria saw 1/5 of the sun • Angular size of the moon and sun is ~0.5°, so θ=1/10° • Knowing the distance D between the two cities, he derived the distance to the moon from D=Dm θ

The Dark Ages

Aristotle, 384-322BC

Ptolemy, 85-165AD

The Church

•

Aristotle said “If the stars affixed to the celestial sphere are not centered on the earth, and the earth is rotating around the sun, we should see some displacement of the stars. We don’t see this, so unless the stars are ridiculously far away, the earth is the center of the universe!”

•

Aristotle also said “The natural state of a body is to be at rest, and only the presence of a force or impulse would move it. Therefore a heavy body should fall faster than a light one, because it would have a greater pull towards Earth.”

•

Aristotle and Ptolemy prevailed with their Geocentric model of the universe until the 16th century. Everything was OK except for those darned “wanderers” or planets.

History

The Copernican model Nicolaus Copernicus discovered that the seemingly complicated motions of the planets could be explained if the sun was stationary and the planets Mercury, Venus, Earth, Mars, Jupiter and Saturn orbited about the sun.

400th Anniversary of Galileo

Galileo

Used new invention, telescope, to observe jupiter

If moons orbit jupiter, why not earth orbit the sun?

History

Newton’s laws

• In 1687 Newton published Philosophiae Naturalis Principia

Mathematica • A body at rest stays at rest, a body in motion stays in motion with

the same velocity unless acted upon by an external force.

Distance to Planets Parallax distance S un E arth M ercu ry !

Venus transit, APOD July 20, 2004

Relative scales of the solar system Planet

Period (years)

Approx. Radius (a.u.)

Earth

1.0

1.0

Mercury

0.241

0.39

Venus

0.615

0.72

Mars

1.881

1.5

Jupiter

11.86

5.2

Physics 312 - Lecture 3

– p.9/12

Venus transit, Crow Observatory, June 5, 2012

Summary

•

Eclipses imply earth is round, sun and moon have about the same angular size

•

Length of shadows at different latitudes was used to measure radius of the earth

•

Distance to Venus and Mercury were originally determined by parallax during transits using the known radius of the earth as the baseline

•

Radar ranging of the planets (Mars) is now more precise

•

Knowing the distance to one planet determines value of 1 AU, and scale of entire solar system: 1 AU = 1.496×108 km = 93 million miles

•

Can determine mass of the sun to be 1.989×1030 kg from Newton’s law

GM! m mv 2 = r r2

v = 2πr/P

Parallax Distance 1 AU 1 AU d= ≈ tan p p

1 pc d≈ p”

• By definition, a star at a distance of one parsec (1 pc) will have a parallax angle of one arcsec (1”)

• 1 pc = 3.08568025 x 1018 cm • 1 pc = 3.26163626 ly • Nearest star, Proxima Centauri,

has a parallax angle of 0.77” and a distance of 1.3 pc or 4.2 ly

5 References 6 External links

Nearest Stars

List Designation

#

1

System

Star

Solar System

Sun

Alpha Centauri (Rigil Kentaurus; Toliman)

Declination [2]

variable: the Sun travels along the ecliptic

Distance[4] Light-years (±err)

Additional references

180°

0.000015

has 8 planets

11.09[2] 15.53[2]

14h 29m 43.0s

!62° 40! 46" 0.768 87(0 29)"[5][6] 4.2421(16)

# Centauri A (HD 128620)

2

G2V [2]

0.01[2]

14h 39m 36.5s

!60° 50! 02"

# Centauri B (HD 128621)

2

K1V [2]

1.34[2]

5.71[2]

14h 39m 35.1s

!60° 50! 14"

4

M4.0Ve

9.53[2]

13.22[2]

17h 57m 48.5s

+04° 41! 36" 0.546 98(1 00)"[5][6] 5.9630(109)

5 M6.0V[2] 13.44[2] 16.55[2]

10h 56m 29.2s

+07° 00! 53" 0.419 10(2 10)"[5]

6 M2.0V[2]

11h 03m 20.2s

+35° 58! 12" 0.393 42(0 70)"[5][6] 8.2905(148)

06h 45m 08.9s

!16° 42! 58" 0.380 02(1 28)"[5][6] 8.5828(289)

01h 39m 01.3s

!17° 57! 01" 0.373 70(2 70)"[5]

Wolf 359 (CN Leonis)

4

Lalande 21185 (BD+36°2147)

Luyten 726-8

Right ascension[2]

Parallax[2][3] Arcseconds(±err)

M5.5Ve

3

6

G2V [2] !26.74 [2] 4.85[2]

Epoch J2000.0

1

Barnard's Star (BD+04°3561a)

5

Apparent Absolute magnitude magnitude (m V ) (M V )

Proxima Centauri (V645 Centauri)

2

Sirius (# Canis Majoris)

Star #

Stellar class

4.38[2]

0.747 23(1 17)"[5][8] 4.3650(68)

7.47[2]

10.44[2]

7

A1V [2]

!1.46 [2]

1.42[2]

Sirius B

7

DA2 [2]

8.44[2]

11.34[2]

Luyten 726-8 A (BL Ceti)

9

M5.5Ve

12.54[2] 15.40[2]

Luyten 726-8 B (UV Ceti)

10

M6.0Ve

12.99[2] 15.85[2]

Sirius A

7.7825(390)

8.7280(631)

7

Ross 154 (V1216 Sagittarii)

11

M3.5Ve

10.43[2] 13.07[2]

18h 49m 49.4s

!23° 50! 10" 0.336 90(1 78)"[5][6] 9.6813(512)

8

Ross 248 (HH Andromedae)

12

M5.5Ve

12.29[2] 14.79[2]

23h 41m 54.7s

+44° 10! 30" 0.316 00(1 10)"[5]

3.73[2]

6.19[2]

03h

7.34[2]

9.75[2]

23h 05m 52.0s

!35° 51! 11" 0.303 64(0 87)"[5][6] 10.742(31)

15 M4.0Vn

11.13[2] 13.51[2]

11h 47m 44.4s

+00° 48! 16" 0.298 72(1 35)"[5][6] 10.919(49)

16

13.33[2] 15.64[2]

9

Epsilon Eridani (BD!09°697)

13

K2V [2]

10

Lacaille 9352 (CD!36°15693)

14

M1.5Ve

11

Ross 128 (FI Virginis)

12

EZ Aquarii (GJ 866, Luyten 789-6)

EZ Aquarii A

M5.0Ve

32m

55.8s

!09° 27! 30" 0.309 99(0

79)"[5][6]

[2] 15.58[2] M? !15°encyclopedia 18! 07" 0.289 50(4 40)"[5] 13.27 22h 38m the 33.4s free List 16 of nearest stars - Wikipedia,

EZ Aquarii B EZ Aquarii C

[7]

16

M?

14.03[2] 16.34[2]

10.322(36) has two 10.522(27) proposed planets

11.266(171)

Magnitude Scale M=1

M=2

M=3

M=4

M=5 M=6

• Hipparchus (followed by Ptolemy) created a catalog of Brighter Star (to be measured) Reference Star

about 1000 stars that were grouped into six magnitude groups. Ptolemy called the brightest stars first magnitude or m=1, the second brightest m=2 and so on.

• In the early 19th century, William Herschel (born in Aperture D1 Aperture D2

Hanover, Germany 1738 - built massive 48” reflector and 20’ refractor) devised a naked-eye method to make quantitative measurements of magnitude

• Herschel’s method consisted of viewing a reference star (with a stopped-down telescope) and an unknown with a star (with an identical telescope). When the aperture was adjusted so that the apparent magnitudes were the same, the apparent magnitude could be determined:

F1 · πD12 /4 = F2 · πD22 /4

F1 /F2 = (D2 /D1 )2

Stellar Magnitudes • In 1856, Pogson made more precise measurements verifying Hershell’s result that a first magnitude star is about 100 times brighter than a 6th magnitude star.

• Pogson formalized the system, the ratio of brightness of two stars with apparent

magnitude differing by 1, was defined to be exactly 1001/5=2.512, now known as the Pogson ratio.

• Pogson’s scale was originally fixed by assigning Polaris a magnitude of 2.

When Polaris

was found to be variable, Vega became the standard reference with m=0.

• Some examples:

The sun has m=-26.73, the full moon m=-12.6, maximum brightness of Jupiter m=-2.94, brightest star Sirius m=-1.47, Vega m=0.03, Andromeda galaxy m=3.44

• The absolute magnitude M of a star is defined so that it is the same as the apparent

magnitude for stars at a distance of 10 pc (typical for nearby stars). From your textbook: 1001/5 = 100.4 = 2.512

F1 /F2 = 100.4(m2 −m1 ) Fabs /Frel = r2 /(10 pc)2 = 100.4(m−M ) M = m + 5 − 5 log r

Solar Luminosity

L⊙ = 3.836 × 1033 erg s−1

L⊙ 6 −1 −2 Radient flux at Earth : F = = 1.36 × 10 erg s cm 4πd2

Solar constant : F = 1.36 kW m−2

Standard Candles • Understanding stellar structure is a key to finding “standard candles” • Even if we can’t determine the absolute luminosity of a particular type of star, can use as a standard candle if we:

✴ know the luminosity is a constant (not usually the case) ✴ know the dependence of luminosity on some parameter (e.g., color, spectral lines, period of oscillation, etc.) ✴ have a calibration through a parallax distance to some representative members of this class

• One motivation of understanding stellar structure, is to be able to use stars to map out the large scale structure and evolution of the universe.

• A classic example of a standard candle is a “Cepheid variable”.

For these stars the luminosity varies periodically, with the absolute luminosity tightly correlated with the period. Getting parallax distances for a few of these, calibrates the use of Cepheids as standard candles for distance measurements.

Tully Fisher Relation

• In 1977 astronomers Brent Tully and Richard Fisher determined an

empirical relationship between intrinsic luminosity and rotation velocity of spiral galaxies.

• Rotation velocities are readily measured by Doppler shifts of spectral lines.

• As for all standard candles, must calibrate some representative objects by another distance measure (e.g., Cephied variables) and can then use the apparent brightness and inverse square law to determine distance

Distances in our Galaxy • Transitions in relative spins of electrons and protons in

neutral hydrogen give rise to 21 cm (1420 MHz) radiation

p

e

➱

p

e

• Molecular clouds containing hydrogen, CO, etc. rotate around GC in Keplerian orbits.

• Doppler shifted 21 cm line gives line-of-sight velocity -

can use Kepler’s laws to reconstruct distribution of matter in galaxy, distances, enclosed mass (Dark Matter!)

Sun

• Association of galactic objects (e.g., supernova remnants, pulsars) with molecular clouds can give a crude distance (sometimes the only distance) to galactic objects.

Distance http://abyss.uoregon.edu/~js/ast123/lectures/lec13.html

• Parallax to kpc • Spectroscopic Parallax • Cepheids to 20 Mpc • Tully-Fisher relation to > 100 Mpc • Type Ia supernovae to > 1000 Mpc

Standard Candles and Redshift • If we know the intrinsic luminosity of stars, and measure their brightness we can measure distance

• Often hard to know the intrinsic luminosity.

Objects for which we have some basis of calibrating intrinsic luminosity are known as standard candles and can be used for measuring distance (or rigorously luminosity distance)

• Hubble observed more distant (fainter) galaxies appeared to recede more quickly. Now standard candles can be used to calibrate the redshift-distance relationship (Hubble’s law) to map the most distant universe.

Edwin Hubble, born 1989 in Marshfield, MO!

Redshift /Velocity Relationship A modern Hubble diagram

λobs = λrest

B A R(t)

�

1 + vr /c 1 − vr /c

λobs − λrest Redshift : z ≡ λrest � 1 + vr /c z= −1 1 − vr /c 1 + vr /c 1 + 2vr /c + (vr /c)2 v/c � 1 ⇒ = ≈ 1 + 2vr /c 2 1 − vr /c 1 − (vr /c) [1 + 2vr /c]

1/2

vr z≈ c

H0 d vr = c c

1 ≈ 1 + 2vr /c = 1 + vr /c 2

What are Stars made of?

Spectra

• Spectroscopy is the key to understanding the composition of stars, stellar structure, physical parameters of stars

Spectral Classification

Annie Canon observing in 1895. At the time Wellesley students observed using a 4-inch Browning telescope that could be set up on the north or south porch of College Hall.

•

Harvard group expanded upon the spectral classification subdividing into 15 types labeled A through O according to the strength of the Balmer lines.

•

M.N. Saha, an Indian physicist showed how, for a given temperature, one could calculate the most likely energy level at which an atom's electrons could be found.

•

Cecilia Payne used Saha's work to show how the strength of hydrogen lines depends on temperature.

•

Annie Canon, who actually led the Harvard classification project, reorganized the classification scheme based on this work, rearranging the classes from hot to cold – O, B, A, F, G, K, M.

Spectral Classification

Spectral Classification

that relate the two systems:

CGS Units

1 Ba = 1 g/(cm·s2 ) = 10-3 kg/(10-2 m·s2 ) = 10-1 kg/(m·s 2 ) = 10-1 Pa.

Definitions and conversion factors of CGS units in mechanics CGS unit

CGS unit abbreviation

Symbol

length, position

L, x

mass

m

gram

g

1/1000 of kilogram = 10!3 kg

time

t

second

s

1 second

=1s

velocity

v

centimetre per second

cm/s

cm/s

= 10!2 m/s

force

F

dyne

dyn

g cm / s 2

= 10!5 N

energy

E

erg

erg

g cm2 / s 2

= 10!7 J

power

P

erg per second

erg/s

g cm2 / s 3

= 10!7 W

pressure

p

barye

Ba

g / (cm s 2 )

= 10!1 Pa

dynamic viscosity

μ

poise

P

g / (cm s)

= 10!1 Pa·s

wavenumber

k

kayser

cm!1

cm!1

= 100 m !1

centimetre

cm

Definition

Equivalent in SI units

Quantity

1/100 of metre

= 10!2 m

Centimeter gram second (cgs) system of units - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Centimetre_gram_second_system_of_units

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