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## Red Giants and White Dwarfs

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### Red Giants and White Dwarfs

A Field Guide to Stars

The Solar Neighborhood

Milky Way Galaxy

100 billion stars

Volume of nearly 100,000 light years across

Orbits Galactic Center- 25,000 light years from Earth

Measuring Parsecs

- Parallax decreases as distance increases
- Distance (in parsecs)= 1

parallax (in arc seconds)

Parallax=0.5”

1/0.5= 2pc

Parallax=0.1” 1/0.1=10pc

One Parsec≈3.3 Light years

Luminosity and apparent brightness

Luminosity is intrinsic also called the absolute brightness

We see apparent brightness

Another Inverse Square Law

Leaving a star, light travels through imaginary spheres of increasing radius surrounding the source.

Inverse Square

- Doubling the distance from a star makes it appear 22, or 4 times dimmer.
- Tripling makes it look 32, or 9 times more dim.
- Luminosity also affects brightness.
- Doubling the luminosity also doubles the energy crossing any spherical shell surrounding the star.
- This doubles the apparent brightness.
- The apparent brightness of a star is directly proportional to the star’s luminosity and inversely proportional to the square of its distance.

Determining Luminosity

- Two things are needed
- Determine apparent brightness
- Star’s distance
- Magnitude Scale
- Second century Greek astronomer Hipparchus
- Classified into six groups

The Magnitude Scale

- The use of telescopes that could measure energy shows two important facts
- The 1-6 magnitude range spans a magnitude of 100 in apparent brightness
- Hipparchus used his eyes

Modern Magnitude Scale

- Define a change of 5 in magnitude to correspond to exactly a factor of 100
- 1-6 or 7-2..
- Numbers in Hipparchus’s ranking are apparent magnitudes
- Scale is no longer limited to whole numbers
- Magnitudes outside of the 1-6 range are allowed

Absolute Magnitude

- Measures apparent brightness when the star is seen at its actual distance from the sun
- Absolute magnitude is apparent magnitude from 10 parsecs from the observer
- Inverse Square (again)
- Star @ 100pc “moved” to 10pc
- Distance decreases by a factor of 10
- Apparent brightness increases 102 or 100 times
- Its apparent magnitude would decrease by 5

More on the Magnitude Scale

- Sun’s absolute magnitude is 4.83
- Since an increase in brightness by a factor of 100 corresponds to a reduction in a star’s magnitude by 5 units, a star with a luminosity 100 times that of the Sun has an absolute magnitude of
- 4.83-5=-0.17
- A star with .01 Solar luminosity has an absolute magnitude of
- 4.83+5=9.83

More on the Magnitude Scale

- We can fill in the gaps if we realize 1 magnitude corresponds to a factor of 1001/5≈2.512, 2 magnitudes to 1002/5≈6.310 and so on.
- A factor of 10 in brightness corresponds to 2.5 magnitudes.

Luminosity Conversion Chart

Calculate the luminosity (in solar units) of a star having absolute magnitude of M. The star’s absolute magnitude differs from the Sun by (M-4.83) magnitudes, So the luminosity, L, differs from the solar luminosity by a factor of:

100 -(M-4.83)/5 or

L(solar units)= 10–((M-4.83)/2.5)

From appendix 3:

MSun=4.83, has L=100=1

Sirius A with M=1.45, has L=101.35=22 Solar Units

More on the Magnitude Scale

- Barnard’s Star with M=13.24, has L=10-3.5 = 4.3x10-4 Solar Units.
- Betelgeuse has M= -5.14 and L=9,700 suns…

Converting Luminosity to Absolute Magnitude

- Invert the previous formula
- M=4.83-2.5 log10L
- Vega: L=50
- M=4.83-2.5 log(50)
- M=0.58
- Eridani: M=0.3
- M=4.83-2.5 log(0.3)
- M=6.2

Apparent Luminosity, Absolute Magnitude, and Distance

- m=M+5 log (D/10pc)
- Or
- D= 10pc X 10((m-M)/5)
- Knowing the difference m-M between apparent and absolute magnitudes is equivalent to the objects distance from us.

Apparent Luminosity, Absolute Magnitude, and Distance

- The star Rigel:
- m=0.18
- D=240 pc
- M= 0.18-5log(24)=-6.7

Stellar Temperatures

- Color and the Blackbody Curve
- Measure the apparent brightness at several different frequencies
- Match observations to appropriate blackbody curve

Blackbody Curves

B and V filters admit different amounts of light for objects of different temperatures.

Spectral Classification

- Between 1880 and 1920 stellar spectra was collected
- No firm theories on how the lines were produced
- Stars were classified by their hydrogen-line intensities
- Now are classified as O, B, A, F, G, K, and M.

Spectral Classification

- Astronomers further divided each letter into 10 subdivisions
- Our sun is a G2 (cooler than a G1, but hotter than a G3)
- Vega: A0
- Barnard’s Star: M5
- Betelgeuse: M2

Direct and Indirect Measurements

- With distance known and angular diameter measured, we can calculate actual radius.
- 130pc and angular diameter of up to 0.045”
- Betelgeuse’s maximum radius is 630 times that of the Sun. (Betelgeuse is a variable star).
- Most stars are too distant or too small to be measured directly

Radiation Laws

- Stefan Boltzmann Law
- Energy emitted per unit area per unit time increases as the fourth power of the star’s surface temperature.
- Large bodies radiate more energy than do small bodies at the same temperature
- Luminosity α radius2 X temperature4
- Radius-Luminosity-Temperature relationship:
- Knowledge of a star’s luminosity and temperature can yield an estimate of the star’s radius

Estimating Stellar Radii

- Stefan Boltzmann law: F=σT4
- Area of a sphere: A=4πR2
- Luminosity α radius2 X temperature4
- So
- Luminosity= 4π σ R2 T4
- Or
- Luminosityα radius2 X temperature4

Estimating Stellar Radii

- Use solar units
- L (in solar luminosities)= 3.9x1026W
- R (in solar radii)= 696,000 Km
- T (solar temperature)= 5800K
- We can eliminate the constant 4π σ and rewrite the equation as
- L (in solar units)= R2 (in solar radii) x T4 (in units of 5,800K)

Estimating Stellar Radii

- L (in solar units)= R2 (in solar radii) x T4 (in units of 5,800K)
- To compute the radius, we change the formula to
- R=√L/T2
- Aldebaran
- Surface Temperature: 4000K
- Luminosity: 1.3x1023W
- So the luminosity is 330 times the Sun and temperature is 4,000/5,800= 0.69
- R=√330/0.69
- R=18/0.48
- R=39 solar radii

Estimating Stellar Radii

- Canopus, the second brightest star in the southern sky
- Apparent magnitude of -0.62
- Parallax of 0.0104”
- Distance (pc)= 1/ parallax
- 1/0.0104= 96pc
- M=m-5log(dist/10pc)
- M=-0.62-5log(9.6)
- M=-5.5

Estimating Stellar Radii

- M=-5.5
- L= 10 –(M-4.83)/2.5
- L= 10-(-5.5-4.83)/2.5
- L=10 -(-4.132)
- L≈ 14,000
- Canopus spectral type is an F0: implying a surface temperature of 7,400 K or 1.3 solar temperature
- L=R2xT4
- R=√L/T2
- R=√14,000/1.69
- R≈70 solar Radii

Giants and Dwarfs

- Giants are any star whose radii are between 10 and 100 solar radii.
- Aldebaran is red in color, so it is classified as a Red Giant.
- Stars ranging up to 1000 solar radii are known as supergiant
- Betelgeuse is a supergiant

Giants and Dwarfs

- Sirius B
- T= 27,000 K (4.5)
- L= 1025W (0.025)
- R=√0.025/4.52
- R=0.007 solar radii
- A dwarf is any star whose radius is comparable to or smaller than the Sun (including the Sun)
- Because any 27,000 K object glows blue-white, Sirius B is a white dwarf.

Hertzsprung-Russell Diagram

Relationship exists between stellar temperature and luminosity

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