Chapter 11 Stars

• Properties of Stars• Classifying Stars

– Hertzsprung-Russel (H-R) Diagram

• Star Clusters– Open and Globular

Clusters

Properties of Stars

• Mass – The single most important property that determines other properties of the star.

• Luminosity – The total amount of energy (light) that a star emits into space.

• Temperature – surface temperature, closely related to the luminosity and color of the star.

• Spectral type – closely related to the surface temperature

• Size – together with temperature determine the luminosity

What can we measure directly?

The Easy Ones: • Apparent brightness: a well-calibrated detector.• Temperature: spectroscopy• Spectral type: spectroscopy

The Hard ones:• Distance: stellar parallax, but the stars are so farrrrr

away…• Size: The stars are so far away. Their small angular

size makes it really difficult to be measured directly. • Mass: Newton’s version of Kepler’s Third Law

– Need to find the right targets

The Apparent BrightnessApparent brightness • The brightness of the a star as it appears to our

eyes (or detectors). • It depends on both the luminosity AND distance

between the star and the Earth. The apparent brightness of a star is related to its

luminosity and distance by the formula:

• The total energy in this cone is fixed…• At a larger distance from the star, the same

amount of energy is spread into a larger area. Thus, the apparent brightness of a star is lower if we are further away from it.

2distance)(4π

luminositybrightnessapparent

The Magnitude SystemApparent magnitude describes the relative brightness of objects as they

appears in sky.A difference of 5 magnitudes is equivalent to a factor of 100 difference in apparent brightness. 1st magnitude star is 100 times brighter than a 6th magnitude star. A difference of one magnitude is a factor of 2.51 difference in brightness. The larger the magnitude, the fainter the object Objects with negative magnitude appear brighter than objects with

positive apparent magnitude.

Apparent magnitude mv of selected objects :• The brightest star in the in night time sky, Sirius, is mv = -1.4• The Sun: mv = -27• The full Moon is -13• Maximum brightness of

– Venus: mv = -4.7– Mars: mv = -2.9– Jupiter: mv = -2.8

• Large Magellantic Cloud: mv = 0.9• Andromeda galaxy: mv = 4.3• Faintest star visible to human eyes: mv = 6

The Absolute Magnitude

A star’s absolute magnitude Mv is the apparent magnitude it would have if it were at a distance of 10 parsecs (32.6 light-years) from Earth.

– The Sun’s absolute magnitude is Mv = 4.8

– Sirius: Mv = +1.4

– Betelgeuse: Mv = -5.1

• Apparent magnitude tells us nothing about the luminosity of the objects, but it tell us how difficult it is to see the objects in the sky.

• Absolute magnitude, on the other hand, is directly related to the luminosity of the object. But it does not tell us how bright they appear in the sky.

Astronomical Distance

Measuring the Temperature of StarsEverything with a temperature emit thermal radiation. We can measure the temperature of the stars or any object by studying the shape of their overall spectra.

Black BodyAn idealized perfect light absorber that absorbs all the photons that strikes it (no reflection). It re-emits the absorbed energy through thermal radiation, with a spectrum characterized by the blackbody spectrum.

• The shape of the blackbody spectrum is always the same, independent of its temperature.

• The peak position (in wavelength) of the blackbody spectrum depends only on the temperature, independent of the blackbody’s composition, or size, etc.

Spectral Type of Stars• Spectral type is closely related to temperature

Spectral Type and Temperature

The spectral features of the stars are closely related to the surface temperature of the star because the formation of ionized atoms, the excitation state of the atoms, and the existence of molecules in the stellar atmosphere strongly depends on the temperature • High temperature

Ionized atoms

• Medium temperature Neutral atoms

• Low temperature Molecules

Determination of DistanceStellar Parallax Knowledge of the distance to the

stars is crucial for our determination of the luminosity of stars…

• Current technology allows us to determine the distance accurately to within a few hundred light-years.

• Hipparcos mission (European Space Agency) measured the stellar parallax of roughly 100,000 stars with precision of a few milli-arcseconds. So, it can measure distance of star up to 1,000 light-years away…

Simulation of Stellar Parallax…

Astronomical Distance Units• Light-year:

– The distance light travels (in vacuum) in one year.– one light-year is 10 trillion (1013) km

• Parsec: parallax & arcsecond– One parsec: the distance to an object with a parallax angle

of 1 arcsecond.– One parsec equals to 3.26 light-year.– kiloparsecs: 1,000 parsecs.– megaparsecs: 1,000,000 parsec.

Absolute Magnitude

Determination of Stellar Mass Mass is the single most important property of a star. But it is also

difficult to measure…

The most dependable method we have for measuring the mass of distant stars is Newton’s version of Kepler’s Third Law of orbital motion (Problem 33, Chapter 4).

Recall that

So, if we can find 1. two stars (binary star system) orbiting each other, and 2. if we can measure their

• rotational period p, and • semi-axis a of the orbit,

then we can determine their masses.

3

21

22

)(

4a

mmGp

Binary Star SystemsBinary star systems are formed by two stars that are gravitationally bounded,

and they orbit each other.

About 50% of the stars are in binary star system. There are three categories of binary star systems: • Visual Binary: a pair of stars that we can see distinctly (with a

telescope) as the stars orbit each other.• Eclipsing Binary: is a pair of stars that orbit in the plane of our line of

sight. The stars are not resolved, but we can see the effects of the stars blocking each other in their combined light-curve.

• Spectroscopic Binary: in some binary system, we cannot see the two stars, nor can we see their light curve changes, but we can see the motion of the stars from Doppler effect measurement of the spectra.

BCenter of

mass

A

True Binary Star System

Binary Star Systems Two stars appearing close to each other in the sky do not necessarily means

that they are a binary system.

If A and B are not gravitationally bounded with each other, then, although they may appears to be very close in the sky, they do not constitute a binary system!

A

B

B Line-of-Sight

A

Visual Binary – Sirius

Sirius (in constellation Canis Major) is the brightest star in the night-time sky (magnitude -1.4). It is a visual binary system. Sirius A (the larger of the two) is a main sequence star with spectral type A0, and Sirius B is a white dwarf.

Sirius A & B time sequence

Hubble Space Telescope image of Sirius

White Dwarf

Eclipsing Binary About 50% of the stars are in binary star system. There are three

categories of binary star systems: • Eclipsing Binary: is a pair of stars that orbit in the plane of

our line of sight, (measuring the time curve)

Animations source: http://en.wikipedia.org/wiki/Spectroscopic_binary

Algol – Eclipsing Binary

Algol (the demon star) is in the constellation of Perseus. • Algol A: main sequence

star, more massive.• Algol B: subgiant, less

massive.

The Algol Paradox: Why is the more massive Algol A evolve slower than the less massive Algol B? (Next chapter).

Spectroscopic Binary• Sometimes only the spectrum from one star is

seen, the other star is too dim.• Sometimes two sets of spectra can be seen at the

same time• Sometimes more than two sets of spectra can be

seen– Mizar is a visual binary system in the

constellation of Big Dipper.– Each ‘star’ in the visual binary system is also

a spectroscopic binary!

Eclipsing Binary and Stellar Mass Measurements

Among the three types of binary star systems, the eclipsing binary system is most important for the determination of stellar mass, because– Determination of the stellar mass requires knowledge of the orbital

period and distance (in real distance unit, not in angular separation).

– Orbital period is easy to measure, but distance between the stars is difficult to determine.

• For visual binary, we need to know the distance from Earth to the stars before we can determine the separation between the stars in the binary system.

• For spectroscopic binary, we can calculate the separation between the stars if we know their orbital speed. However, we can only determine the line-of-sight speed of the binary system from Doppler measurement. If the orbits are tilted with respect to our line-of-sight, then we under estimate the orbital speed.

– If an eclipsing binary is also a spectroscopic binary, then we know its true orbital speed, and can determine the separation between the two stars. Then, the masses of the stars can be determined!

Luminosity To directly measure the luminosity of a star (let’s say, the Sun), we will

need to surround the Sun completely with detectors, which is impossible.• We can infer the luminosity of the Sun if we know

– the distance to the star, and – the star’s apparent brightness

Further more, we need to assume that• the star emits energy uniformly in all direction…

Then we can calculate its luminosity by the formula:brightnessapparent 4πLuminosity 2 d

The total area of the sphere with a radius of r is 4d2

d

Quiz: Which Star Has Higher Luminosity?

Star Apparent Brightness B Distance d

A 10 1

B 1 10

• The apparent brightness decrease as d 2

• The brightness of star A is 10 × 1 = 10 • The brightness of star B is 1 × 102 = 100

if observed at distance 1 Star B is 10 times more luminous than A!

The photons contained in box A are spread into an area 4 times as large in box Y which is twice the distance from the star as X.

X

Y

Luminosity of Selected Stars

Star Distance

[ly]

Spectral

Type

Luminosity

[L/Lsun]

Prosima Centauri

4.2 M5.5 0.0006

Bernard’s Star 6.0 M4 0.005

Gliese 725 A 11.4 M3 0.02

Centauri B 4.4 K0 0.53

Sun 0.000016 G2 1.0

Centauri A 4.4 G2 1.6

Sirius A 8.6 A1 26.0

Vega 25 A0 60

Achernar 144 B5 3,600

Betelgeuse 423 M2 38,000

Deneb 2500 A2 170,000

Luminosity and Distance — Chicken and Egg

brightnessapparent 4π

luminosity

d

Most of the time, we need measurement of distance to calculate the luminosity. Howver, if we can determine the luminosity of an object with other methods (independent of distance measurement, such as the luminosity of supernovae), then we can derive the distance to the object from measurement of their apparent brightness.

Direct Measurement of the Size of the Stars

Except for the Sun, all the stars in the sky are very far away, and their angular sizes (the size of the star as it appears to observers on Earth, not the physical size) are all very small. Although the theoretical resolving power of modern large telescopes (such as the Keck telescope with 10-meter aperture) is about 0.01 arcseconds in the visible wavelength, it is difficult to realize the full resolution of the large telescopes due to atmospheric seeing effects.

Interferometry have directly measure the angular size of stars. Direct measurement by interferometry can achieve about 0.01 arcseconds angular resolution.• The angular size of Betelgeuse was first observed using interferometry in

1921…0.051 arcseconds.• R Doradus (in constellation Dorado in the southern hemisphere) is the

star with the largest observed angular size: 0.057 arcseconds.– 0.057 arcseconds is equal to 0.000016 degrees!

If we know the angular size and the distance of a star, we can derive its physical size

Size of star = angular size [radian] distance

Optical Interferometry The technique of combining images from multiple telescopes to obtain

very high resolution images…Recall that the resolving power of telescopes is fundamentally limited by the size of the telescope. – However, it is not necessary to build a single telescope with

sufficient size to achieve the required resolution. Theoretically, multiple small telescopes separated by a large distance can achieve the same resolution of that of a single large telescope.

– The two 10-meter Keck Telescopes at Mauna Kea are separated by 85 meters distance. When they are used together as an interferometer, the theoretical resolution is equivalent to that of a single 85-meter diameter telescope.

– Interferometry is routinely used for observation in radio frequency.

Betelgeuse and R Doradus

• The physical size of Betelgeuse (a red supergiant) is roughly 500 times the size of the Sun, or 4.6 AU (radius of 2.3 AU, or 345 million km).

• The size of R Doradus (a red giant) is 370 times the size of the Sun, or 3.4 AU (radius of 1.7 AU). – If R Doradus or Betelgeuse

are placed at the center of our solar system, then their surface would extends beyond the orbit of Mars (1.5 AU, or 225 million km).

Image of hot spots on Betelgeuse from http://www.mrao.cam.ac.uk/telescopes/coast/betel.html using interferometric technique.

Giants and Supergiants

Since the stars are so far away, we can only directly measure the angular size of just about 10 stars by interferometric technique so far. However,if we know the luminosity (from apparent brightness and distance measurements) and the temperature of the stars, then we can calculate their physical size:

Assuming that stars are blackbody• The energy output of a unit surface area on the surface of the star is

determined by its temperature (Stefan-Boltzman Law)• The total energy output (luminosity) therefore depends on the

temperature and its total surface area, which is related to its size.

where r is the radius of the star.• We can then calculate the size of the star by

Indirect Determination of the Size of Stars

424luminosity σTπr

44

luminosity

π σ Tr

Properties of Stars: Summary

• Mass range: 0.08 Msun to 100 Msun

• Luminosity range: 0.00001 Lsun to 1,000,000 Lsun

• Size range: 0.01 Rsun for white dwarf to 1,000 Rsun for supergiants.

• Temperature range: 3,000 K for M star to 40,000 for O stars.

• Properties of Stars• Classifying Stars

– Spectral Type and Luminosity Class

– Hertzsprung-Russel (H-R) DiagramMain Sequence StarsGiants and SupergiantsWhite Dwarf

• Star Clusters– Open and Globular Clusters

Clues to Relationships Between the Properties of Stars

General trends of the stars…

• Most of the very brightest stars are reddish in color.

• If we ignore those relatively few bright red stars, there’s a general trend to the luminosities and colors among all the rest of the stars:

– The brighter ones are white with a little bit of blue tint,

– the more modest ones are similar to our Sun in color with a yellowish white tint, and

– the dimmest ones are barely visible specks of red.

Hertzsprung-Russell

Diagram Since there appears to be a

strong correlation between luminosity and color (temperature), we put all the stars on a Luminosity – Temperature plot, and this is what it looks like:

Properties of Stars shown in the H-R Diagram: 1. Luminosity (log scale).2. Temperature and spectral

type 3. Size4. Mass of the main

sequence5. Lifetime

Sizes scale 1 Rsun 10 Rsun 100 Rsun 1000 Rsun

Hertzsprung-Russell

DiagramNotice that…• Temperature scale decreases

from left to right.• The scale of luminosity is in

power of 10 (log scale).• Mass increases from lower

right to upper left• Size increases from lower left

to upper right.

Sizes scale 1 Rsun 10 Rsun 100 Rsun 1000 Rsun

Classification of Stars in H-R

Diagram1. The Main Sequence stars

• healthy stars, fusing hydrogen in the core.

• High-mass, high-luminosity, high-temperature, and short-lived stars on the upper-left-hand corner

• Low-mass, low-luminosity, low-temperature, and long-lived stars on the lower-right-hand corner

2. The Supergiants, 3. The Giants,

• Supergiants and giants are dying stars, fusing helium and heavier elements.

4. The White Dwarfs. • dead stars, exposed core of

dead main-sequence stars.

Sizes scale 1 Rsun 10 Rsun 100 Rsun 1000 Rsun

Classification of Stars

Full classification of stars includes both spectral type and luminosity class: – Spectral type: OBAFGKM – Luminosity Class in descending order:

• I: Supergiants • II: Bright giants• III: Giants• IV: Subgiants• V: Main-sequence stars

– The full classification of a star includes both a spectral type and a luminosity class:

• The Sun is a G2 V• Proxima Centauri is M5 V• Betelgeuse is M2 I• Sirius A: A1 V• Sirius B: DA2 V