Temperature
What is a
Hertzsprung-
Russell diagram?
An H-R diagram plots the luminosity and temperature of stars

Most stars fall somewhere on the main sequence of the H-R diagram

Large radius
Stars with lower
T and higher L than mainsequence stars must have larger radii:
and

Small radius
Stars with higher
T and lower L than mainsequence stars must have smaller radii:

Spectral lines of supergiants, giants, and mainsequence stars
Figure 8.9: Differences in widths and strengths of spectral lines distinguish the spectra of supergiants, giants, and main-sequence stars. (Adapted
from H. A. Abt, A. B. Meinel, W. W. Morgan, and J. W. Tapscott, An Atlas of
Low-Dispersion Grating Stellar Spectra, Kitt Peak National Observatory,
1968)
Fig. 8-9, p. 141

A star’s full classification includes spectral type
(line identities) and luminosity class (line shapes, related to the size of the star):
I - supergiant
II - bright giant
III - giant
IV - subgiant
V - main sequence
Examples: Sun - G2 V
Sirius - A1 V
Proxima Centauri - M5.5 V
Betelgeuse - M2 I

Luminosity classes
The approximate location of the luminosity classes on the H–R diagram.
Fig. 8-10, p. 142

A few stars are big enough to resolve in the new telescopes.
Sizes for these can now be calculated directly.
Betelguese is 600X larger than the Sun.

Using the luminosity and the temperature of a star the area and hence the size of a star can be determined.
𝐿 = 𝜎𝐴𝑇 4 = 4𝜋𝜎𝑟 2 𝑇 4 𝑟 =
𝐿
4𝜋𝜎𝑇 4

Polaris has just about the same spectral type
(and thus surface temperature) as our sun, but it is 10,000 times brighter than our sun.
𝐿 = 4𝜋𝜎𝑟 2 𝑇 4
Thus, Polaris is 100 times larger than the sun .

Temperature
H-R diagram depicts:
Temperature
Color
Spectral Type
Luminosity
Radius

A
B
D
C
Temperature

A
B
D
C
A
Temperature

A
B
D
C
Which star is the most luminous?
Temperature

A
B
D
C
Which star is the most luminous?
C
Temperature

A
B
C
Which star is a mainsequence star?
D
Temperature

A
B
D
C
Which star is a mainsequence star?
D
Temperature

A
B
C
Which star has the largest radius?
D
Temperature

A
B
D
C
Which star has the largest radius?
C
Temperature

Main-sequence stars are fusing hydrogen into helium in their cores like the Sun
Luminous mainsequence stars are hot (blue)
Less luminous ones are cooler
(yellow or red)

Low-mass stars
High-mass stars
Mass measurements of main-sequence stars show that the hot, blue stars are much more massive than the cool, red ones

Low-mass stars
High-mass stars
The mass of a normal, hydrogenburning star determines its luminosity and spectral type!

Core pressure and temperature of a higher-mass star need to be larger in order to balance gravity
Higher core temperature boosts fusion rate, leading to larger luminosity

For stars
Some HR diagrams use the magnitude instead of the luminosity. While it looks like a linear scale remember it’s really a log scale.

If we know a star’s absolute magnitude or luminosity, we can infer its distance by comparing absolute and apparent magnitudes:
Distance Modulus
= m
V
– M
V
= -5 + 5 log
10
(d [pc])
Distance in units of parsec
Equivalent: d = 10 (mV – MV + 5)/5 pc

Stellar Distances. From parallax outward each new method is calibrated by data from the earlier method. The methods overlap.

Luminosity: from brightness and distance
10 -4 L
Sun
- 10 6 L
Sun
Temperature: from color and spectral type
3,000 K - 50,000 K
Mass: from period (p) and average separation (a) of binary-star orbit
0.08 M
Sun
- 100 M
Sun

Luminosity: from brightness and distance
(0.08 M
Sun
) 10 -4 L
Sun
- 10 6 L
Sun (100
M
Sun
)
Temperature: from color and spectral type
(0.08 M
Sun
)
3,000 K - 50,000 K (100
M
Sun
)
Mass: from period (p) and average separation (a) of binary-star orbit
0.08 M
Sun
- 100 M
Sun

Notice the scales are log scales.
The diagonals are lines of constant radius, from the radius-luminosity-temperature relationship.

p. 151

There should be nothing special about our local neighborhood. It therefore is useful as a random sample of stars in the galaxy.
The larger our sample size the better our conclusions. So a complete sample out to greater distances gives a more accurate picture of the galaxy as a whole.

Faint, red dwarfs (low mass) are the most common stars.
Bright, hot, blue mainsequence stars (highmass) are very rare.
Giants and supergiants are extremely rare.

Sun’s life expectancy: 10 billion years
Until core hydrogen
(10% - 15% of total) is used up
Life expectancy of 10 M
Sun star:
10 times as much fuel, uses it 10 4 times as fast
10 million years ~ 10 billion years x 10 / 10 4
Life expectancy of 0.1 M
Sun star:
0.1 times as much fuel, uses it 0.01 times as fast
100 billion years ~ 10 billion years x 0.1 / 0.01

Stellar Population Distribution Explained
Faint, red dwarfs
(low mass) are the most common stars.
Long lived
Bright, hot, blue main-sequence stars (high-mass) are very rare. Short lived
Giants and supergiants are extremely rare.
Passing through a stage.

High Mass:
High Luminosity
Short-Lived
Large Radius
Blue
Low Mass:
Low Luminosity
Long-Lived
Small Radius
Red

• Stellar properties depend on both mass and age: those that have finished fusing H to He in their cores are no longer on the main sequence
• All stars become larger and redder after exhausting their core hydrogen: giants and supergiants
• The core of most stars ends up small and white after fusion has ceased: white dwarfs

A
Temperature
B
C
D
Which star is most like our
Sun?

A
Temperature
B
D
Which star is most like our Sun?
B
C

A
Temperature
B
C
D
Which of these stars will have changed the least 10 billion years from now?

A
Temperature
B
D
Which of these stars will have changed the least 10 billion years from now?
C
C

A
Temperature
B
C
D
Which of these stars can be no more than
10 million years old?

A
Temperature
B
D
Which of these stars can be no more than
10 million years old?
A
C

• Any star that varies significantly in brightness with time is called a variable star
• Some stars vary in brightness because they cannot achieve proper balance between power welling up from the core and power radiated from the surface
• Such a star alternately expands and contracts, varying in brightness as it tries to find a balance

• The light curve of this pulsating variable star shows that its brightness alternately rises and falls over a
50-day period

• Most pulsating variable stars inhabit an instability
strip on the H-R diagram
• The most luminous ones are known as
Cepheid variables

• Cepheid variable’s magnitude will vary between 0.5-2 magnitudes over a period from days to months.
• They have been show to have a period of variability that depends on luminosity.
• This allows Cepheid variables to be used as a standard candle to measure distance.

• The accepted explanation for the pulsation of
Cepheids is called the Eddington valve
• Doubly ionized helium is more opaque than singly ionized helium.

• The more helium is heated, the more ionized it becomes.
• At the dimmest part of a Cepheid's cycle, the ionized gas in the outer layers of the star is opaque, and so is heated by the star's radiation
• Due to the increased temperature the fusion rate in the core increase resulting in an increase in radiation pressure causing the star to begin to expand.

• As it expands, the ionized helium cools, and becomes less ionized and therefore more transparent, allowing the radiation to escape cooling the star and slowing the fusion rate.
• Then the expansion stops, and reverses due to the star's gravitational attraction.
• The process then repeats.

• What is a Hertzsprung-Russell diagram?
– An H-R diagram plots stellar luminosity of stars versus surface temperature (or color or spectral type)
• What is the significance of the main sequence?
– Normal stars that fuse H to He in their cores fall on the main sequence of an H-R diagram
– A star’s mass determines its position along the main sequence (high-mass: luminous and blue; low-mass: faint and red)

• What are giants, supergiants, and white dwarfs?
– All stars become larger and redder after core hydrogen burning is exhausted: giants and supergiants
– Most stars end up as tiny white dwarfs after fusion has ceased
• Why do the properties of some stars vary?
– Some stars fail to achieve balance between power generated in the core and power radiated from the surface

• Our goals for learning
• What are the two types of star clusters?
• How do we measure the age of a star cluster?

Open cluster: A few thousand loosely packed stars, typically found is galactic disk, and typically young in age <5 billion yrs.

Globular cluster: Up to a million or more stars in a dense ball bound together by gravity, most found in the galactic halo with some in the disk, some of the oldest objects found in the galaxy.

Massive blue stars die first, followed by white, yellow, orange, and red stars

Mainsequence turnoff point of a cluster tells us its age

Main-sequence turnoff
Pleiades now has no stars with life expectancy less than around 100 million years

Detailed modeling of the oldest globular clusters reveals that they are about 13 billion years old

• What are the two types of star clusters?
– Open clusters are loosely packed and contain up to a few thousand stars
– Globular clusters are densely packed and contain hundreds of thousands of stars
• How do we measure the age of a star cluster?
– A star cluster’s age roughly equals the life expectancy of its most massive stars still on the main sequence