Lecture 2

Observables: luminosities & distances

University of Naples Federico II, Academic Year 2011-2012

Istituzioni di Astrofisica, read by prof. Massimo Capaccioli

Learning outcomes

The student will see:

• how stars are named;

• a recapitulation of concepts of photometry;

• what are stellar magnitudes and colors, and hoe they are measured;

• some of the basic photometric systems of magnitudes;

• how cosmic distances are measured: parallaxes , photometric distance

indicators, Hubble law;

• how to define absolute magnitudes.

• that magnitudes and colors of stars correlate in the so called HR diagram.

Shedir (α Cas)

A stellar field: use “figures” to commit it to memory

Constellations are usually just prospectical

Orion in 3D andin projection

Naked-eye stars carry many names, from familiar to hopelessly obscure.Vega, the 4-th brightest star in the sky, has over 40 different names!

Star have been named in all epochs. Names are now assigned by astronomical organizations on behalf of the International Astronomical Union(IAU, the worldwide organization of professional research astronomers). No private organizations have the rights to name stars.

There are many Greeknames, but most come from Arab(in the Middle Ages astronomers of Arabian countries adopted Ptolemy’s Greek constellations). These names were passed back to Latin, often in a highly corrupted form, also mistakenly transferred from one star to another.

Names of stars & constellations

To familiarize with the sky, download this software: http://www.stellarium.org/

In 1603 a German astronomer and lawyer, Johannes Bayer, named stars coupling lower case Greek letter to constellation namesmore or less in order of brightness: so the brightest star in a constellation is “Alpha”, the second “Beta”, and so on.

To the Greek letter name is appended the Latin possessive form of the constellation name: thus the brightest star in Lyra, Vega(an Arabic proper name), becomes Alpha Lyrae.

Bayer followed the lower-case Greek alphabet (24 letters) with lower-case Roman letters, then with upper case Roman letters. These are rarely used nowadays.

Greek letter names

In the XVIII century John Flamsteedlisted the stars by position within constellation boundaries. Serial numbers applied later gave the star's relative location from West to East within the constellation. For example, 1 Lyrae would be the Western-most star in Lyra.

To include yet more stars, constellations are now dropped and the stars are named according to position, generally Right Ascension, or angle to the east of the Vernal Equinox.

For a tutorial on the Astronomical Coordinate Systems see App. Lect. 2

Star cataloguesfall into two broad categories: general and special interest.

Cf. the WORD file:Stars and Constellations.doc, for a complete list of the names (+ mening) of the brightest stars and of all the constellations.

More on stellar names ….

John Flamsteed First Royal Astronomer

(1646-1719)

The appearance of stars

Betelgeuseis a red giant, and one of the largest stars known.

Rigel is one of the brightest stars in the sky; blue-white in colour.

Betelgeuse = α Orionis(hand of the central one)

Rigel = β Orionis(foot of the central one)

Bayer’s Uranometria

Stars exhibit a large variety of brightnesses and colors.

photometry

radiation specific intensi

Let's begin with a recapitulation of some basic concepts of .

The or is defined as the carried b

ty brigy the

htness amount of light of

energywavelength b en etwedEλ

( ) 1 2 2 1

and in the through the ins

timearea solid angle

cos

cos

, erg s cm st

ide a

e z

:

r H

d dtdA d

dE dI d

dtdA d

dE d I d dtdA d

I

λλ

λ λ

ν

λ λ λ

λλθ

λ λ θ

ν − − − −

+Ω

=Ω

= Ω

Ω

x

y

z

dA φ dφ

θ

sind d dθ θ φΩ =

dθ

( )I Tλ

Photometry in a nutshell: radiation intensity

2

5

Max Planck in 1901 radiation black body absorption coefficient idenThe formula derived by for the from a

, . . a body with that's why isat blall wa ackvelengths ( ), is

tica

2( )

1

l to 1

hc

kT

hcB T

i

e

e

λλ

λ

λ=

−

( )( )

3

2

2 / 2

0 0 0

or

The emitted by a i

s just:

We will do this int

total energ

(how ?)2

( )

1

y rate bla

cos

cos

ck body

si

n

h

kT

A

hB T

c e

dE dI dA d d L d

dt

L B d dA d d

ν ν

λλ λ

π π

λφ θ λ

ν

λ θ λ λ

λ θ θ θ φ∞

= = =

=

−

= Ω =

= ∫ ∫ ∫ ∫

4Stefegrat an-Boion later ltzmann laand find the :w .L A Tσ=

cos

dE d

dI d

dA dtλ

λλλθ

=Ω

Max Planck (1858-1947)

x

y

z

dA φ dφ

θ

sind d dθ θ φΩ =

dθ

( )I Tλ

Radiation intensity of a black body

0

In general, depends on .

The is defined to be the radiated in ( over all

directi

solid a

on

mean intensity average intngles ):

ensity

all directions

1

4

1

4

I

d

I I d I

i.e.

λ

π

λ λ λφ θπ π =

Ω

= Ω =∫ ∫2

0

isotropic directio

If is ( . . independent of and );then:

n

This is true of a black b

si

y

n

od .

d d

I

J I

i

I

e

π

λ

λ λ λ

θ θ φ

φ θ

=

≡ =

∫

x

y

z

dA φ dφ

θ

sind d dθ θ φΩ =

dθ

( )I Tλ

Mean intensity

2 2

00 0 0

Remember that:

sin cos 4d d dπ π π

π

φ θ φ

θ θ φ θ φ π= = =

= − =∫ ∫ ∫

Evaluate the energy density associated with radiationc

:os

dE dI d

dtdA dλ

λλλθ

=Ω

z

θ

dA

dL

Energy density

4 4

0

16 3 4

blackbody raIn the special case of :

diation

ra

cos

1 1 4

diation

where is t

4 4

7.566 10 J m K h

e .constant

c dt dA

dL dA

u B

dE du d I d d I

c

d T aTc c

c

a

λλ λ λ

λ

θ

π

λ πλ λ

σλ∞

− − −

= = Ω

×

= =

×

=

=

=

∫ ∫

∫

Evaluate the associated with the radiat energy density cos

ion:dE d

I ddtdA d

λλ

λλθ

=Ω

z

θdA

dL

Energy density

The is the with wavelength between and that passes thro

radiative flux net energy unugh a

i

cos

dE dI d

dtdA

d

dλ

λλλθ

λ λ λ+

=Ω

, co

t area unit time coming from all directions

isotropic radiation no ne

in a .

For there is (an equal amount passes

t flux through

the unit area in op

sradf d d I dλ λλ λ θ= Ω∫

posing directions).

Radiative net flux

θ

z

z

of carries a :

Light therefore exerts a :

Consider a with hitting a

A photon energy momentum

radiation pressure

beam of radiation

energy s

1

E p E c

F dp dt dE dtP

A A c A

dE dλ λ

=

= = =

( ) ( )2,

urfaceangle

at an :

cos1 1cosrad

dA

dE dt ddP I d d

c dA cλ

λ λ

θ

θ λλ θ= = Ω

Radiation pressure

cos

dE dI d

dtdA dλ

λλλθ

=Ω

θ

z

z

( ) ( )

4

8 2 44

8

1 4

3

4 5.67 10 Wm K293 K

2.997

room temperature

A9 10 m s

t ,

radP u u Tc

u

σ

− − −

= =

× ×= =

×6

6 12

477 18

6

61 Pa = 10 a

.

and the is: ( ).

How

ever, at , t

pressure

prehe is:

5.57 10 J m

1.8 10 Pa 1.8 10 atm

1010 K 1.3 10

2 tissure mes l

9arger,

t

3

2.3 1 0 t

o

m

a

r

P

T

P

−

− −3

− −

×

= × = ×

= = ×

= × m.

Black body radiation pressure

cos

dE dI d

dtdA dλ

λλλθ

=Ω

( )

,

2,

Mean intensity (sometimes written ):

Energy density:

Radiative f

1

4

1 4

cos

1cos

lux:

Radiation pressure:

rad

rad

J I I d

u d I d d Ic c

f d d I d

P I d dc

λ λ λ

λ λ λ

λ λ

λ λ

π

πλ λ

λ λ θ

λ θ

= Ω

= Ω =

= Ω

= Ω

∫

∫

∫

∫

Recap of the formulae

Observable properties of stars

In total, 3 independent quantities to be chased observationally.

[ ]

[ ]

33

33 1

1.989 10

Basic parameters to compare theory with observations are:

, usually in Solar masses ( )

, usually in Solar luminosities

Mass

Luminosity

g

3.846 10 erg ( )

It is t

he tota

s

M M

L L −

= ×

= ×

⊙

⊙

[ ]0

10

l energy radiated per second, . . power (in W):

, usually in Solar rad 6.955 Radius 10 equator

Effective

ii ( at t

temperat

he )

c

m

u

L L d

R R

i e λ λ∞

=

= ×

∫

⊙

[ ] ( )( )

This is the temperature of a of the same radius as the star that would radiate the same amount of e

(see laternergy

5778

(no ma

re

bltter what

the spectrum is

ack body )e eT T K

RL

= °⊙

2 4 8 2 4

). Thus : , wher(see later) 4 5.6 e . 7 10 W m KeL R Tπ σ σ − − −= = ×

Apparent brightness of stars

Star nameRelative

brightnessDistance [light years]

Sirius -α CaM 1 8.5

Canopus -α Car 0.49 98

Alpha Cen 0.23 4.2

Vega -α Lyr 0.24 26

Arcturus -α Boo 0.25 36

Capella -α Aur 0.24 45

The apparent brightnessof stars depends on both: their intrinsic luminosity, their distancefrom the observer, the intervention of some sort of extinction.

Their colorsare independent of distance.

Why this is so?

The five brightest stars

The five nearest stars

Star nameRelative

brightnessDistance [light years]

Proxima Cen 0.0000063 4.2

Alpha Cen 0.23 4.2

Barbard’s star 0.000040 5.9

Wolf 359 0.000001 7.5

Lalande 21185 0.00025 8.1

In the following we will consider • brightnessand• distanceof the stars.

Apparent magnitudes

magnitude systemlogarithmic scale relative assigned zero pThe expresses fluxes in a given waveband on a

to an :

oint

n

Note the

2.5log

ref

refref

ff

fm m

f

λ

λλ

λ

− = −

: it means that have .

The scale is chosen so that a factor corresponds

egative signbrighter object

100 in brightness 5 magnitude

to match the ancient H

s

i

to (

pparc

s lower magnitud

os-Ptolemy m

es

agnitude scale).

Point sorces

Apparent magnitudes: examples

Object Apparent visual mag

Sun −26.5Full moon −12.5Venus −4.0Jupiter −3.0Sirius −1.4Polaris 2.0Eye limit 6.0Pluto 15Reasonable telescope limit (8-m telescope, 4-h integr.)

28

Deepest image ever taken (Hubble UDF)

29

9

(29 6)/2.5 46/5

At 29, this reaches more than 10 times fainte

The faintest (deepest) telescope image taken so far is the

r than we can see with the naked eye. 10

HST Ultra-Deep Field.

10

m

−

=

= ≈ 910

2.5logrefref

fm m

fλ

λ

− = −

30 HST, Keck limit

6 Naked eye limit

25 Palomar Hale telescope limit

18 1 m telescope

10 Binocular limit9.5 Barnard’s star

2.5 Polaris limit 0 α Cen limit

−6 Venus at brightest

−12.5 Full Moon

−26.8 Sun

Ap

par

ent v

isu

al m

agn

itud

e

α Cen

Barnard’s star

Venus

Polaris

Apparent magnitudes: examples

Stars of different apparent magnitudesThe summer triangle

α Cygni

α Lyrae

α Aquilae

Standard star

Actual measure of a monochromatic flux

( )( )

0Actually, the is quite different from the

measure relative to a band centered at , returned by the detector of the

observing equipment

flux outside

.

The

the atmosphere

mai n causesof this chan

f

f

λλ λ

0

ge are a consequence of:

1. the ,

2. the of the various elements in the observing equipment.

Atmospheric extinction can be taken

atmospheric extinction

filtering

care by a proper observational

b

r

y

st

f

ategy.

plane

parallel lay ho

Let's mod

w good?

el the atmosphere by

( ).

ers

extinction along

( ) ( )

T the line of sighe is:

,

wher

ht

extie the nction coefficient

x

dfk h dx

f x

k

λλ

λ

= −

0 0

0

it depends on

time too, indeed sec = a

depends on and (

). Since ( ):

Integration along the visual gives:

ir massec

( )sec sec . ( )

sec

altitu

s

(

de

h h

h

x h z

dfk h z dh z d

f h

dfz d f h

f

z

λλ λ

λ

λλ λ

λ

λ

τ

τ∞ ∞

=

= − = −

= − →∫ ∫

0 0

sec ( )) ( ) e

( ) optical

, where:

, is d( ept) tsec hhe .

z

h h

f

d k h z dh

λτλ

λ λ λτ τ

− ∞

∞ ∞

= ∞

∞ = =∫ ∫

Zenith

dh dhsecz

x

zground at h0

above sea level

zenithal distance

Atmospheric extinction

[ ] [ ]( ) ( )

sec ( )0

0

out atm.

( ) ( ) e

2.5log ( ) 2.5log ( ) +2.5 ( ) log esec

sec ( )

z

obs

f h f

f h f z

m m k z k X z

λτλ λ

λ λ λ

λ λλ λ

τ

− ∞= ∞− = − ∞ ∞

− = − = −

0.0 1.0 2.0

∆m

secz

Bouguer line

magnitude lossat Zenith

extinction coefficient generalized air mass

( )

( ) ( )sin 90 sin sin cos cos cos

Observe the same star at

all along the night; the

for a source at seen at

different zenithal

distances zenithal

distance

sidereal time lat from i tude is :

.

Then e

,

s

s

z t

t

φ δ φ δ α

α δφ

− = + −

xtrapolate to to derive the

c Evaluat

zero air

e the soorrection urces of

mas

err

s

. ors.

Atmospheric extinction

Actual measure of a monocromatic flux

( )

The effects of ( . ., glass filters, reflectivities or transmission properties of the optics, sensitivity of the detector) may

all the filters

transfer function be summed up into a

which acts on the

e g

T λ( )

( )( ) ( )

( )

0

0

0

0

of the object, , in such a way that:

, where is typically the center of t

spectral energy distribution

broad bandhe

defined by . Idea zlly, is ero eve but in the intrywhere e

f

T f d

f

T d

TT

λ

λ λ λλ λ

λ λ

∞

∞=∫

∫

( ) ( )( )

2

1

2

1

1 2

0

2 1 2 1

rval where it is unity.

If so, , the is , and: band wid2 th

f d

f

d

λ

λλ

λ

λ λ

λ λλ λ λ λ λ λ λ

λ

÷

= + ∆ = + =∫

∫

Artificial lights at night

Artificial lights at night

The sky brightness at Zenit

Hal

f ab

sorp

tion

alti

tud

e (k

m)

Wavelength (Å)

The atmospheric absorption

Spectra of the night sky and of Sun

( ) ( )

( )0

0

2.5log

T f d

m q

T dλ λ

λ λ λ

λ λ

∞

∞

= − +

∫

∫

Photometric bands

( )T λ

Several have been defined, mostly in an empirical way. The following slides review som

photometric magnitudee of the more popular

sys on

temses.

Wavelenght [Ångstrom]

log

Flu

x

5

4

3

2

1

4000 5000 6000 7000

filter passband

The Johnson-Morgan UBVRI magnitude systemDefinition: α Lyr (Vega) has V = 0.03 and all colors equal to zero.

The flux of α Lyr calibrated on an absolute scale by Hayes and Lathan (1975): 3500 Jansky at 5556Å.

W.W. Morgan(1906-1994)

Band U B V R I

λc/nm 365 445 551 658 806

Width/nm 66 94 88 138 149

The Gunn griz magnitude systemsOriginally defined in terms of photoelectric detectors(Thuan & Gunn 1976; Wade et al. 1979); now used primarily with CCDs(Schneider et al. 1983; Schild 1984).

Definition: the star BD+17°4708, a subdwarf F6 star with B−V = 0.43, is defined to have colors equal to zero.

Jim Gunn (1938-)

Trinh Thuan(1948-)

Symbol λe(µm) ∆λ(µm) (FWHM)

Passband (µm)

u 0.353 0.04 0.320 ÷ 0.380

v 0.398 0.04 0.360 ÷ 0.440

g 0.493 0.07 0.462 ÷ 0.568

r 0.655 0.09 0.600 ÷ 0.755

i 0.820 0.13 0.700 ÷ 0.876

AB magnitude system

This magnitude system is defined in such a way that, when monochromatic flux f is measured in erg sec−1 cm− 2 Hz− 1:

m(AB) = −2.5 log( f ) − 48.60

where the value of the constant is selected to define m(AB) = V for a flat-spectrum source.

In this system, an object with constant flux per unit frequency interval has zero color.

John Beverly Oke(1928-2004)

Band name

Central λλλλ[mm]

Bandwidth[mm]

Flux of Vega[Jy]

U 0.37 0.066 1780

B 0.45 0.094 4000

V 0.55 0.088 3600

R 0.66 0.14 3060

I 0.81 0.15 2420

J 1.25 0.21 1570

H 1.65 0.31 1020

K 2.20 0.39 636

( ) ( )( )

,

0,

2.5log 2.5log

2.5logVegam f f

f mλ λ λ

λ λ

= − += − +

Cfr. http://www.astro.utoronto.ca/~patton/astro/mags.html#conversions

Imagine a source which has a constant flux of 10 Jy at all frequencies.

What is its magnitude in the U band? And in the V and K bands?

N.B.: the flux unit or Jansky (symbol Jy) is a non-SI unit of electromagnetic flux density equivalent to 10−26 W m−2 Hz−1.

Apparent magnitudes: exercise

Karl Gunther Jansky (1905-1950)

Colors

1 2

1 2 1 2

1 2

1

The difference between two monocromatic magnitudes and , with , is named

logaritmic ratio of thecolor.

It is the at and :

( ) ( )( )

fluxes

( ),

( ) (

m mm m

m m

λ λλ λ λ λ

λ λ

λ λ

− <

− 12

2

( )) 2.5log

( )

f

f

λλ

= −

Colorspassbands cutting off spectral energy distributions

Cool star Hot star

larger (B−V)

smaller (B−V)

What is the B−V color of a source that has a flux proportional to λ−4 ?

Colors: exercise

( )4

0.5512.5log 2.5log 10log 10 log 1.238 0.93 mag

0.445V V

B B

fB V

f

λλ

− = − = = = =

Band U B V R I

λc/nm 365 445 551 658 806

Width/nm 66 94 88 138 149

Note that the color does depend (strongly) on absorption but not on distance.

Why ? Study scattering!

basic unit Astronomical Unit circular

In the cosmic distance scale, the is the = radius of of a point orbit sidereama l ss o yearrbiting about the

semimajo

(AU)

r ax S

isun in one

of Earth'

s orb≈

2 3

it

Kepler's third law

How do we measure this

.

Relative distances of planets from Sun can be determined from : .

which, given , for Earth and Mars, writes:

?

Mars

Mar

P aP P

P

P

⊕

⊕

∝

2 3

8

.

1 AU = 1.49598×10 km.

s Mars

a

a⊕

=

Unit of distance: the Astronomical Unit (AU)

360° relative to distant stars

A measure of the Astronomical Unitsynodic period time lapsed

two consicute alIf is the of a body X orbiting around the Sun (between of X with Earth & Sun in the same order),

then it is easy to see that:

ignments

see f

1 1( r

1

X

X X

S

S P P⊕= −

2 32 3

).

From Kepler's third law: , .

At the time of the , the distance of from isclos

est approach X :, covered

om Ear

in aEarth

th the motion of X

AU

- t

XX

X X

X X

P a Pa

P a Pi.e.

d a a

⊕ ⊕

⊕

⊕

= =

∆ = ime by a .So, send a at the epoch of the and measure the time it take

radar si

s to com

gnalradar signal to Venus clos

21

2 2 2

est

e back, approa

ch

1

2 2

X

Venus Venus

Venus

t d c

t

t a a c a a ac

t a P P

⊕ ⊕ ⊕

⊕

∆ = ∆

∆

∆ = − = −

∆ = ( )2 3

where the only unknown is

11

.c

a

⊕

⊕

−

The “parallax” is the apparent shift in position of a nearby star, relative to background stars, as Earth moves around the Sun in its orbit.

This defines the unit 1 parsec = 206265 AU = 3.0857×1013 km ~ 3.2616 ly.

1 AU

p

d

second

s of a

rc

parsec

parallax in

distanxe to star in s

1

d ppd

==

=

Stellar Parallax: 1 pc ≈ 3.3 ly

Measuring Parallax The star with the largest parallax is Proxima Cen,

with p = 0".7723±0".0024

(from the Hipparcos Catalogue, cfr. webpage).

Compute its distance ? d [pc] =1/p"

These small angles are very difficult to measure from ground; the atmosphere tends to blur images on scales of ~1 arcsec.

It is possible to measure smaller parallax angles but only down to ~ 0.02 arcsec(corresponding to a distance of 1/0.02 = 50 pc).

Until recently, accurate parallaxes were only available for a few 102 very nearby stars.

Comparison of ground-based picture of the GC M4 with an HST Wide-Field Planetary Camera 2 image showing white dwarfs.

Proxima Centauri

Since Earth is ~8 kpc from Galactic Center, it is clear that this method is only useful for stars in the immediate Solar neighborhood.

The best ladder: Hipparcos

The Hipparcos satellite (launched in 1989; cfr. webpage) collected parallax data from space, for over 3 years.

120,000 stars with 0”.001 precision astrometry; over 1 million stars with 0”.03; the distance limit corresponding to 0”.001 is 1 kpc (1000 pc).

region about the Sun explored by Hipparcos

Hipparcos’ range at Milky Way scaleNotice how small is the region about the Sun where Hipparcos measured parallaxes

Since for the nearest stars d > 1 pc, we will measure p < 1 arcsec;

e.g.at d = 100 pc, p = 0.01 arcsec.

Telescopes on groundhave a resolution~ 1";

HSThas a resolution 0.05" ⇒even so, distances are difficult to gauge!

Hipparcos(High Precision Parallax Collecting)

satellite measured 105 bright stars with δp ~ 0.001"

⇒ confident distances for stars with d < 150 pc.

Hence ~105 stars with well measured parallax distances.

HST

Hipparcos

Limits of the parallax

Parallax: summary

1. A fundamental geometricmeasurement of distance.

2. It can be measured directly.

3. It is limited to nearby stars.

4. It is used to calibrateother, less direct distance indicators,

thus constructing a cosmic ladder.

5. Ultimately, even our estimates of distances to the most remote

galaxies rests on a reliable measure of parallax to the nearest

stars.

For small angles: p = 1/d [rad]

d = 1/p" [parsecs]if is in arc seconds

1AUp p

d

•

d

1 AU

Measured energy flux f depends on distanceto star (inverse square law):

f = L /4πd2

where L is the flux at a standard distance (equal for any observer).

Hence, only if d is known, L can be determined determined, and viceversa.

Photometric distances are based on distance indicators, i.e. on classes of recognizable objects/phenomena of know andstable luminosity L.

Photometric distance indicators are the key tools to fill the gap between the range of applicability of the parallaxesand that where theHubble flow of the expanding Universe dominates over peculiar motions.(i.e. where the redshift z = ∆λ/λ 0 is proportional to the distance via the Hubble constant: cz = H0d). The link is needed to derive a value for the Hubble constant, which establishes the rate of expansion of the Universe here and now and which cannot be computed from the theory.

Photometric distance indicators are essential in the study of stars, and some stars are extraordinary distance indicators.

(but absorption & scattering)

Photometric distance

0 1 2 3 4 5 6 7 80

50

100

100×

∆d/d

(err

ore

%)

log(d[pc])

0 67 670 6700 cz[km/s]

Trigonometric parallaxes

Hipparcos satellite

Hubble law

Distance indicators

to relate the standard meter with the Hubble constant H0

Pleiades

Crab Nebula Galactic Center

Milky Way

M31

Virgo Cluster

Coma Cluster

Proxima Centauri

?base

rad

Rd

α=

radbase 2

rad

d Rα

α∆

∆ =

cos 0mV H d=

peccos pec 1m

VV V V V

V

= − = −

The zoo of distance indicators & their ranges

1 10 100 1000 104 105 106 107 108 109 1010 parsec

Redshift

Supernovae

Tully-Fisher and Fundamental Plane

RR-Lyrae and Cepheid Variables

Miscellaneous Stellar Techniques

Main Sequence Fitting

Proper Motions

Parallax

Gravitationally Lensed Quasars

Sunyaev-Zel’dovich Effect

Alp

ha C

enta

ury

Mag

ella

nic

Clo

uds

M31

Com

a C

lust

er

Virg

o C

lust

erM

101

Typical stars in the Solar neighborhood have velocities ~ 30 km/s.

What is the size of their Doppler shift at 5000 Å ? (Ans.: 0.5 Å)

Steller motions: examples

Mizar

The evolution of the shape of the Big Dipper during 200,000 years

Example of how much stars move (tangentially in this case)

epoch

Absolute magnitudes

2

It is also useful to have a measurement of an

. Since: , you must fix for all stars

intrinsic brightness independent of

the distance

absolute magni

.

The , , is defined to b

4

e the magnitudde

tu

Ld f d

d

M

π=

0

*

*

*

e that a star would have if it were at an . It is easy to show tha10 pc

10 pc5log 5log 5

pc

t:

(

arbitrar

note the

ily fixed

zero poin

distance

ts have cancelled out)

pa

wher [ re

d

dm M

d

d

=

− = − = −

] is the of the star of apparent magnitude .

The value of is kn

actual distance

distance modulus town as the (" " if the magnitudes are corrected for

rueabso

se

rp )

c

.on

ti

m

m M−

Useful to evaluate rapidly the observability of a source.

Example

*

6

5lo

Recall that the deepest exposures ever taken rea

ch

g 5 (@ 1 Mpc) 29.76pc

Calculate the apparent magnitude of the Sun (abs. mag. 4.76) at a distance of 1 Mpc (10 pc).

Dm M m

M

− = − =

→

=

⊙

⊙

lim ( ).

The nearest large galaxy to us is Andromeda (M31), at a distance of about .

Detecting stars like the Sun in other galaxies is there

HST Ultra Deep Fie

fore very difficul

29

0.78 Mpc

t (genera y

ld

ll

m =

impossible for now).

Recap:

The absolute magnitudeM is defined as apparent magnitude of a star placed at a distance of 10 pc (in a clear universe):

m – M = 2.5log(d/10)2 = 5log(d) − 5

where d is the distance in parsec.

Magnitudes are measured in some wavelength band: e.g.UBV.

Comparison with theory is simpler via the , defined as the tha

bolometric magnitudeabsolute magnitude sensitive to all wt a detector ( )

would measure.

We define th

avelengths bolometer

bolometric ce orrec

( )

( )

( )

2

1

1 2

0

to be:

with and

Bolometric luminosity (in solar units) is then:

2.5log 5070 5950

Ang

( ) 2.5 l

.

tion

og

bol V

bol bol

f dBC M M

f d

M M L L

λ

λλ λ

λ λλ λ

∞= − = = =

− = −

∫

∫

⊙ ⊙

Absolute & bolometric magnitudes

Bolometric correctionscorrection to incorporate the entire spectrum

BC = Mbol – MV [mag]

What is the passband of a bolometric magnitude ?

From Allen’s Astrophysical Quantities (4th edition)

OB

AF GK

M

Bolometric corrections for Main-Sequence stars

The color-magnitude diagramM, R, L and Te do not vary independently.

Two major relationships:

The first [L = f(T)] is known as the color-magnitude or CM diagram.

L with TL with M

Betelgeuse

Rigel

The Hertzsprung-Russell diagram

Ejnar Hertzsprung (1873-1967)

Henry N. Russell (1877-1957)

Another name of the CM diagram is the ( ) diagram (1914).

It proved to be the key to unlock the secrets

Hertzsprung

of stellar

-Russel

evolut

l H-R

ion.

The CM diagram

2

2

Precise parallax distances allow us to plot a colour-magnitude ( ) diagram for nearby stars.

Colour

c-m

independent of distance, ratio of fluxe

is since it is a s:

4

4

red red red

blue blue blue

f r L L

f r L L

ππ

= =

(y-axis) requires measurement oAbsolute magnitude

flux dista

f & nce.

CMD for Hipparcos with good distances (< 15% errror), that is for nearby stars.

Types of stars

Intrinsically faint stars are more common than luminous ones.

A Census of the Stars

Faint, red dwarfs (low mass) are the most common stars.

Bright, hot, blue Main Sequence stars (high-mass) are very rare.

Giants and supergiants are extremely rare.

Back to distances: Main Sequence fitting

Stellar clusters: Consist of many stars, densely packed.

For distant clustersit is a very good approximation that all the constituent stars are at the same distancefrom us.

Typical clusters have sizes ~1 pc;

so, d > 10 pcthis assumption introduces an error < 10%.

Therefore, we can plot a color-magnitude diagram using only the apparentmagnitude on the y-axis, and recognizable structure appears.

NGC2437

Apparent magnitudes = absolute magnitudes + distance modulus

We can take advantage of the structure in the HR diagram to determine distances to stellar clusters.

Color is independent of distance, so the vertical offset of the Main Sequence gives you the distance modulusm− M.

Main Sequence fitting

Nearby stars (absolute magnitudes via parallax)

Distant cluster (only apparent magnitudes)

Main Sequence fitting: NGC 2437

Nearby stars (parallax) Distant cluster (apparent magnitudes)

Thus the distance modulus is: This gives a distance to NGC 2

10.7 5log 5 log 3.14.1.4437 of , close to the acce kp pte

dc 1. value p .8 k c

Vm M V M d d− = − = = − =⇒

At a colour of , the absolute magnitude of the MS is

1.0 mag

. 6.8 magV

B V

M

− =

=

In NGC 2437, at the same colour,

17.

5.V =

Instead of fitting at one color, you can fit the whole Main Sequence.

Find the distance modulus that gives you the best overlap.

Main Sequence fitting

Hyades cluster

(B−V)

App

aren

t m

agni

tude

V

The Globular Cluster NGC 104 = 47 Tuc

m−

M =

13

.5 m

ag

47 Tuc MS fitted to Hyades’

… but age also mattersyounger

older

Variable stars

The images above show the same star field at two different times.

One of the stars in the field has changed brightness relative to the other stars.

Can you see which one?

t1 t2

Variable stars

The images above show the same star field at two different times.

One of the stars in the field has changed brightness relative to the other stars.

Can you see which one?

blinking between t1 & t2

Many stars show fluctuations in their brightness with time.

These variations can be characterized by their light curve– a plot of their magnitude as a function of time.

These variations can be periodic, semi-periodic, or irregular.

Variable stars

RR Lyrae in the GC M3

Certain intrinsically variable stars show a remarkably strong correlation between their pulsation period and average luminosity.

2.96(log

Modern calibration of the Cepheid in the ,

yields:

where the period is measured in days, and the magnitude is measured in the

- relation Magellanic C

louds

ba n .

1) 4.9

d

IM P

P

P L

I

= − − −

Variable stars H. Leavitt (1868-1921)

SMC

Instability strip

Classical Cepheidsare not the only type of pulsating variable stars.

There is a narrow stripin the HR diagram where many variable stars lie.

Cepheidsare the brightest variables; however they are also very rare.

Cepheids

RR Lyrae

Pulsating whitedwarfs

W Virginis

RR Lyrae starsRR Lyrae stars(absolute magnitudes MV = +0.59 ± 0.33) are much fainter than Cepheids, but have the advantage that:

1. they almost all have the same luminosity and

2. are more common.

They are easily identifiedby their much shorter periods.

Abs

olut

e M

agni

tude

Period (days)

Log (Period)

Schematic representation

RR Lyrae

RR Lyrae stars have average absolute magnitudes M = +0.6.

How bright are these stars in Andromeda (d = 0.778 Mpc) ?

RR Lyrae stars

The distance ladder: summary

1. Find parallax distances to the nearest stars.

Dedicated satellites are now providing these precise measurements for thousands of stars.

Plot stellar absolute magnitudes as a function of color.

2. Measure fluxes and colors of stars in distant clusters.

Compare with color-magnitude diagram of nearby stars (step 1) and use Main-Sequence fitting method to compute distances.

Identify any variable stars in these clusters. Calibrate a period-luminosity relation for these variables.

3. Measure the periods of bright variable stars in remote parts of the Galaxy, and even in other galaxies.

Use the period-luminosity relation from step 2 to determine the distance.

Note how an error in step 1 follows through all the subsequent steps!

Useful reading