Luminosity and Colourof Stars

Original: Michael Balogh, Univ. Waterloo

Modified by H.L. Malasan for internal use

The physics of stars

A star begins simply as a roughly spherical ball of (mostly) hydrogen gas, responding only to gravity and it’s own pressure.

To understand how this simple system behaves, however, requires an understanding of:1. Fluid mechanics

2. Electromagnetism

3. Thermodynamics

4. Special relativity

5. Chemistry

6. Nuclear physics

7. Quantum mechanicsX-ray ultraviolet infrared radio

Course Outline

Part I Basic properties of stars and electromagnetic radiation

Stellar classification

Measurements of distance, masses, etc.

Part II Chemical composition of stars (interpretation of spectra)

Stellar structure (interiors and atmospheres)

Energy production and transport

Part III Stellar evolution (formation, evolution, and death)

White dwarfs, neutron stars, black holes

The nature of stars

• Stars have a variety of brightnesses and colours

• Betelgeuse is 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

Rigel

Apparent brightness of stars

Star nameRelative brightness

Distance (light years)

Sirius 1 8.5

Canopus 0.49 98

Alpha Centauri 0.23 4.2

Vega 0.24 26

Arcturus 0.25 36

Capella 0.24 45

Star nameRelative brightness

Distance (light years)

Proxima Centauri

0.0000063 4.2

Alpha Centauri

0.23 4.2

Barnard’s star

0.000040 5.9

Wolf 359 0.000001 7.5

Lalande 21185

0.00025 8.1

The apparent brightness of stars depends on both:• their intrinsic luminosity• their distance from us

Their colour is independent of distance

The five brightest stars The five nearest stars

The Astronomical Unit

Astronomical distance scale: Basic unit is the Astronomical Unit (AU), defined as the

semimajor axis of Earth’s orbit

How do we measure this? Relative distances of planets from sun can be determined

from Kepler’s third law:

E.g. given Pearth, Pmars:

32 aP 32

Mars

Earth

Mars

Earth

a

a

P

P

1AU = 1.49597978994×108 km

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

This defines the unit 1 parsec = 206265 AU = 3.09×1013 km ~ 3.26 light years

Parallax

1 AU

p

d

Measuring Parallax

The star with the largest parallax is Proxima Centauri, with p=0.772 arcsec. What is its distance?

A star field with 1” seeing

These small angles are very difficult to measure from the ground; the atmosphere tends to blur images on scales of ~1 arcsec. It is possible to measure parallax angles smaller than this, 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 hundred very nearby stars.

Hipparcos

The Hipparcos satellite (launched 1989) collected parallax data from space, over 3 years 120,000 stars with 0.001 arcsec precision astrometryMore than 1 million stars with 0.03 arcsec precisionThe distance limit corresponding to 0.001 arcsec is 1

kpc (1000 pc). Since the Earth is ~8 kpc from the Galactic centre it is clear

that this method is only useful for stars in the immediate solar neighbourhood.

Parallax: summary

1. A fundamental, geometric measurement of distance

2. Can be measured directly

3. Limited to nearby stars

4. Is used to calibrate other, more indirect distance indicators. Ultimately even our estimates of distances to the most remote galaxies rests on a reliable measure of parallax to the nearest stars

Break

The electromagnetic spectrum

• The Earth’s atmosphere blocks most wavelengths of incident radiation very effectively. It is only transparent to visual light (obviously) and radio wavelengths.

• Observations at other wavelengths have to be made from space.

U B V R I

Different filters transmit light of different wavelengths. Common astronomy filters are named:

Blackbodies

The energy radiated from a surface element dA is given by:

dddAdTBddAdTB sincos)(cos)(

Units of B(T): W/m2/m/sr

Blackbodies

Energy quantization leads to a prediction for the spectrum of blackbody radiation:

1

2)(

4)(

5

2

kT

hc

e

hcTu

cTB

The energy radiated from a surface element dA is given by:

dddAdTBddAdTB sincos)(cos)(

Units of B(T): W/m2/m/sr

Planck’s law

Calculate the luminosity of a spherical blackbody: Each surface element dA emits radiation isotropically

Integrate over sphere (A) and all solid angles ()

2

0

2/

0

sincosA

dddAdBdL

dBA

Properties of blackbody radiation

1. The wavelength at which radiation emission from a blackbody peaks decreases with increasing temperature, as given by Wien’s law:

K cm 290.0max T

424 eTRL 2. The total energy emitted (luminosity) by a

blackbody with area A increases with temperature (Stefan-Boltzmann equation)

This defines the effective temperature of a star with radius R and luminosity L

Examples

The sun has a luminosity L=3.826×1026 W and a radius R=6.96×108 m. What is the effective temperature? At what wavelength is most of the energy radiated?

K cm 290.0max T424 eTRL

Example

Why does the green sun look yellow?

The human eye does not detect all wavelengths of light equally

Examples

Spica is one of the hottest stars in the sky, with an effective temperature 25400 K. The peak of its spectrum is therefore at 114 nm, in the far ultraviolet, well below the limit of human vision.

We can still see it, however, because it emits some light at longer wavelengths

K cm 290.0max T

424 eTRL

Apparent magnitudes

The magnitude system expresses fluxes in a given waveband X, on a relative, logarithmic scale:

Note the negative sign means brighter objects have lower magnitudes

Scale is chosen so that a factor 100 in brightness corresponds to 5 magnitudes (historical)

ref

refXf

fmm log5.2

The magnitude scale

ref

refXf

fmm log5.2

One common system is to measure relative to Vega By definition, Vega has m=0 in all bands. Note this does not mean Vega is

equally bright at all wavelengths!

Setting mref=0 in the equation above gives:

X

XVegaX

mf

ffm

,0

,

log5.2

log5.2log5.2

• Colour is defined as the relative flux between two different

wavebands, usually written as a difference in magnitudes

Apparent magnitudes

Object Apparent

magSun -26.5

Full moon -12.5

Venus -4.0

Jupiter -3.0

Sirius -1.4

Polaris 2.0

Eye limit 6.0

Pluto 15.0

Reasonable telescope limit (8-m telescope, 4 hour integration)

28

Deepest image ever taken

(Hubble UDF)

29

The faintest (deepest) telescope image

taken so far is the Hubble Ultra-Deep

Field. At m=29, this reaches more than

1 billion times fainter than what we can

see with the naked eye.

95/465.2/)629( 101010

ref

refXf

fmm log5.2