Stars come in all sizes, small, large

and even larger.

(2) Effective Temperature (T) - determined from continuous spectrum (blackbody curve), Wien’s Law.

Stellar Properties:

(1) Distance – measure parallax for closer stars

There are 6 important properties of stars.:

(3) Luminosity (L) - determined from apparent magnitude and distance, or from spectrum (luminosity class).

(4) Chemical composition - determined from line spectrum.

(5) Radius (R) - determined from luminosity and temperature, or from distance .

(6) Mass (M) - determined from binary stars

• The distance to a star in parsecs is:1

dp

We can get some accurate distances by Stellar Parallax

One arcsecond = 1’’and, is the angular size of a dime seen from 2 miles or a hair width from 60 feet.

1 parsec = 3.26 light-years = 3.09x1013km

•The nearest star, aside from the Sun, is called Proxima Centauri with a parallax of 0.77 arcsecond. Its distance is therefore: 1.3 pc

d 1

0.771.3pc

Inverse Square Law of Brightness

                                                                                                                  

               The Apparent Brightness of a source is inversely proportional to the square of its distance:

•The apparent brightness of a light source varies inversely with the square of the distance

•Brightness = 1/D2 = 1/D2

2-times Closer or 1/2 distance = 4-times Brighter

2-times Farther = 1/4-times as Bright

Black Body Radiation

A Blackbody is a perfect absorber, an object that absorbs light at all wavelengths, and it heats up.

It is also the perfect radiator: emits at all

wavelengths (continuous spectrum)

characterized by its Temperature.

Energy emitted depends strongly on Temp.

Stars act nearly like Black Bodies.

Radiation Laws

Three Laws Characterize Continuous Spectra

Planck

Wien

Stefan-Bolzman

With these laws we can determine the temperature & other characteristics of stars.

1. The Planck radiation law assumes that the object observed is a perfect radiator and absorber of energy (black body).

2. Stars, although not perfect black bodies, are close enough so that Planck curves are useful descriptions of their radiation.

Wien’s Law for Black Bodies : The peak of a black body curve shifts toward shorter wavelength if the temperature is increased.

Black Body Temperature

• Star B is cooler

• Star B is cooler

• Our Sun

• Our Sun

• A cooler star will have a lower, flatter peak closer to the red end.

• Star A is hotter

• Star A is hotter

According black-body radiation, the spectrum of a hotter star will have a higher, sharper peak closer to the blue end of the spectrum.

Continuum & LinesContinuum & Lines

Real stars usually have a blackbody-like continuous spectrum, upon which absorption lines are superimposed

Hydrogen

Continuum

Absorption Lines

A spectrum can be converted to a trace spectrum.

3

3

2.9 10

2.9 10

X T

T X

Wien’s Law

Is measured in nm which is 910 m

or

The peak wavelength of a Black Body depends upon the Temperature.

The higher the temperature, the shorter the wavelength of the peak radiation.

So, we can get the temperature of a star form its spectrum.

Needs to be in meters (m)

The sun’s max intensity is at a wavelength of about 500nm or

Using Wien’s Law, calculate the sun’s

surface temperature.

32.9 10T X

Problem

9500 10X m

3 92.9 / 500 10T X X 6.0058 10 5800T X K

F

R

It only depends upon the temperature of the object and a constant. It’s the rate of heat flow/sec.

4F TStefan-Boltzmann LawIf you know the temperature of a

Black Body then the total energy emitted from from each square meter, called Energy Flux (F), can be calculated.

This is only for a square meter and stars are different sizes, so to find the total energy, which is called Luminosity, we change the formula to :

2 2 44 4L FA F R R T

Luminosity 4T24 R

Temperature

Surface Area

(how hot)

(how big)

So,Luminosity depends upon Radius(R) & Temperature(T)

Luminosity is, the total amount of energy per second emitted. The Star’s total Wattage!

The area of a sphere is , A=

24 R

So, the total energy emitted by the object each second is called the Luminosity (L).

2 2 44 4L FA F R R T

Brightness: How bright something appears to us,depends on temperature, size, and the distance.

These 2 will put out the same energy per square unit because:

This one is much bigger (R)

So the total L is much more.

Greater Temperature

Greater Luminosity

Same Size

same temperature

L = 4 R2 T4

This formula relates a Star’s Temperature

and Surface Area (its size) to its Luminosity.

It’s more natural to compare an object to a known object. Comparing a star to the Sun would be easier and more helpful, since we know a lot about the Sun and it is a star.

L = 4 R2 T4 looks rather messy

is a constant that you will not have to use.

Let’s get rid of the constants !

2 44

4

L R T

L R T

42

T

T

R

R

L

L

Since 4 and are constants , in the nextformula they cancel each other when

compared to the sun.

Giving

Betelgeuse has a Luminosity of 60,000 L and a surface temperature of 3500 K. Find the radius compared to the Sun.

42

T

T

R

R

L

L 2 460,000 3500

1 1 5800

R

2

2

60,000 .1322

453,857.8

673.7

R

R

R R

Example Problem

Suppose a star is 10 times the Sun’s radius, but only ½ as hot.Find the luminosity of the star compared to the Sun.

2 4L R T

L R T

The star is 6.25 times the Sun’s Luminosity

4

21

10 12 (100)( ) 6.251 1 16

LL

L

The next two formulas are for Main Sequence stars only !

3.5L MM is the mass of the star in solar mass.

L is the Luminosity of the star in solar Luminosity.

Life Time is the approximate life time of a MS star in solar life times.

3.5L M 3.54 128L sunL

What is the Luminosity of a MS star that has a mass 4 times the sun ?

2.5

1Time

M Life

2.5

1T

M

How long can a 4 solar mass MS star live ?

2.5

11/ 32 .031

4T Solar life times

2 10

8

3.1 10 1 10

3.1 10

T X X

T X years

Or, since the sun will live for 10 billion years

Over half of the stars in the sky have stellar companions, bound together by gravity and in orbit around each other.

Visual BinariesVisual Binaries

Optical BinariesOptical Binaries- are chance superpositions, where two stars appear close together but do not actually orbit one another. (Like Mizar & Alcor)

Types of BinariesTypes of Binaries

Physical BinariesPhysical Binaries- where one star orbits another, and each star can be seen in the telescope.

OPTICAL DOUBLES• Not a true binary system

• Stars only appear close together in the sky

• Mizar & Alcor in the Big Dipper

While Alcor and Mizar are Optical Double stars and only appear to be near each other, Mizar is actually a Physical Binary star.

Types of Physical BinariesTypes of Physical Binaries

Eclipsing BinaryEclipsing Binary –(If the angle is good ) two –(If the angle is good ) two stars that regularly eclipse one another stars that regularly eclipse one another causing a periodic variation in brightness.causing a periodic variation in brightness.

Spectroscopic BinarySpectroscopic Binary - two stars that are - two stars that are found to orbit one another through found to orbit one another through observations of the Doppler effect in their observations of the Doppler effect in their spectral lines .spectral lines .

At least half of the stars in the sky are binaries. Eclipsing Binary stars are also referred to as Extrinsic Variable Stars.

Orbits and Masses of Visual BinariesThe primary importance of binaries is that they allow us to measure stellar parameters (especially mass). The center of mass is the location where a fulcrum would be placed to balance the stars on a seesaw.

Masses of Binary starsNewton’s Modification of Kepler’s Law

P must be in years, a in AU

M in solar mass, where Sun = 1

A nearby star Epsilon Eridani has a planet circling the star at a distance of 3.4 AU. The period of the planet is 7.1 years. Find the mass of the star, assuming the mass of the planet to be negligible.

2 31 2

3

2

3

2

)

(3.4)

(7.1)

0.78

star

star

star sun

M M P a

aM

P

M

M M

Problem - Ignoring the mass of one object.

2 31 2

3

2

3

2

( )

( )

(20)( )

(50)

( ) 3.2

sirius siriusB

sirius siriusB

sirius siriusB sun

M M P a

aM M

P

M M

M M M

When dealing with binary stars, the mass of the two stars are similar, and cannot be simply ignored. Sirius b is a white dwarf, and its orbital period around Sirius takes 50 years.If the distance between the the stars is 20 AU, find the mass of the stars.

3.2SiriusA SiriusB SunM M M

But we happen to know that Sirius’ mass 1.99 Msun But we happen to know that Sirius’ mass 1.99 Msun

and so,and so,

Sometimes we might be able to get Sometimes we might be able to get information about one of the stars from the H-information about one of the stars from the H-R diagram to help us determine the mass of R diagram to help us determine the mass of both stars.both stars.

So,So, 1.99 3.2sun SiriusB SunM M M

1.2SiriusB sunM M

Eclipsing BinariesSometimes the orbital plane is lined up so that the stars pass in front of each other as seen from the Earth. Each eclipse will cause the total light from the system to decrease.

The amount of the decrease will depend on how much of each star is covered up (they can have different sizes) and on the surface brightness of each star.

Some binaries are too close together to be resolved, you may still be able to detect the binary through the Doppler shift (in one or both stars). They must be relatively close to each other (short orbital period).

If you can see both stars’ spectrums, you may If you can see both stars’ spectrums, you may be able to use Doppler shifts to measure the be able to use Doppler shifts to measure the radial velocities of both stars. radial velocities of both stars.

This gives you the mass ratio, regardless of the This gives you the mass ratio, regardless of the viewing angle (e.g. nearly face-on, nearly edge-viewing angle (e.g. nearly face-on, nearly edge-on, etc.). This is usually useful informationon, etc.). This is usually useful information..

Spectroscopic Binaries

Thank goodness, my brain is full