Our goals for learning:
• How do we measure stellar luminosities?
• How do we measure stellar temperatures?
• How do we measure stellar masses?

How do we measure stellar luminosities?
And why do we care?
Let’s say we want to find out how far away a star is…
We can’t measure the distance directly
But we can easily measure the brightness
And brightness is related to distance and luminosity

Luminosity and distance are not so easy.

Luminosity and distance are not so easy. However…

if we can relate brightness, luminosity, and distance…

we can calculate any of them if we know the other two.

This is very important and useful.

This is very important and useful. But first…

What is luminosity?

Luminosity:
Amount of power a star radiates
(joules/sec = watts)

Luminosity:
Amount of power a star radiates
(joules/sec = watts)
Example: A100 W light bulb has a luminosity of 100 W

Luminosity:
Amount of power a star radiates
(joules/sec = watts)
This is different from brightness:

Luminosity:
Amount of power a star radiates
(joules/sec = watts)
This is different from brightness:
Amount of starlight that reaches Earth
(watts/square meter)

Thought Question
These two stars have about the same luminosity — which one appears brighter?
A. Alpha Centauri
B. The Sun

Thought Question
These two stars have about the same luminosity — which one appears brighter?
A. Alpha Centauri
B. The Sun

How are luminosity and brightness related?

How are luminosity and brightness related?
•Luminosity passing through each sphere is the same

How are luminosity and brightness related?
•Luminosity passing through each sphere is the same
•But the area increases

How are luminosity and brightness related?
•Luminosity passing through each sphere is the same
•But the area increases
•Area of sphere = 4πR 2

How are luminosity and brightness related?
•Luminosity passing through each sphere is the same
•But the area increases
•Area of sphere = 4πR 2
•Divide luminosity by area to get apparent brightness.

The Inverse Square Law for Light
The relationship between apparent brightness and luminosity depends on distance:

Thought Question
How would the apparent brightness of Alpha
Centauri change if it were three times farther away?
A. It would be only 1/3 as bright.
B. It would be only 1/6 as bright.
C. It would be only 1/9 as bright.
D. It would be three times as bright.

Thought Question
How would the apparent brightness of Alpha
Centauri change if it were three times farther away?
A. It would be only 1/3 as bright.
B. It would be only 1/6 as bright.
C. It would be only 1/9 as bright.
D. It would be three times as bright.

Thought Question
How would the apparent brightness of Alpha
Centauri change if it were three times farther away?
A. It would be only 1/3 as bright.
B. It would be only 1/6 as bright.
C. It would be only 1/9 as bright.
D. It would be three times as bright.

The relationship between apparent brightness and luminosity depends on distance:
Measuring brightness is easy, so if we know how far away a star is, we can calculate its luminosity:

The relationship between apparent brightness and luminosity depends on distance:
Measuring brightness is easy, so if we know how far away a star is, we can calculate its luminosity:

The relationship between apparent brightness and luminosity depends on distance:
Measuring brightness is easy, so if we know how far away a star is, we can calculate its luminosity:
Or if we know its luminosity, we can calculate its distance:

The relationship between apparent brightness and luminosity depends on distance:
Measuring brightness is easy, so if we know how far away a star is, we can calculate its luminosity:
Or if we know its luminosity, we can calculate its distance:

So how far away are these stars?

Introduction to Parallax
Parallax is the apparent shift in position of a nearby object against a background of more distant objects.

Parallax of a Nearby Star
•Apparent positions of the nearest stars shift by only about an arcsecond as Earth orbits the Sun, and the shift is smaller for more distant stars.

Parallax of a Nearby Star
•Apparent positions of the nearest stars shift by only about an arcsecond as Earth orbits the Sun, and the shift is smaller for more distant stars.
•These very small angles explain why the Greeks were unable to detect parallax with their naked eyes.

Parallax of a Nearby Star
•Apparent positions of the nearest stars shift by only about an arcsecond as Earth orbits the Sun, and the shift is smaller for more distant stars.
•These very small angles explain why the Greeks were unable to detect parallax with their naked eyes.
•This inability helped delay the acceptance of the geocentric universe for more than 1500 years.

Parallax Angle as a Function of Distance
The parallax angle depends on distance.

Measuring Parallax Angle
Parallax is measured by comparing snapshots taken at different times and measuring the angular size of the star’s shift in position.

Measuring Parallax Angle
Parallax is measured by comparing snapshots taken at different times and measuring the angular size of the star’s shift in position.

This is a right triangle

That means we can use trigonometry

sin p
 1 d

For small angles: sin p ≈ p

So for small angles: p
 1 d

If you express p in arcseconds and d in parsecs…

p = parallax angle d (in parsecs) =
1 p (in arcseconds) d (in light-years) = 3.26

1 p (in arcseconds)

The apparent brightness of stars varies over a wide range

And there is also a wide range of luminosity

Range of luminosities
Most luminous stars:
10 6 L
Sun
( L
Sun is luminosity of Sun)

Range of luminosities
Most luminous stars:
10 6 L
Sun
Least luminous stars:
10
−4
L
Sun
( L
Sun is luminosity of Sun)

Range of luminosities
This is a wide range

Range of luminosities
This is a wide range
To compress it, the magnitude scale was devised, originally by the
Greeks

Range of luminosities
The Greeks assigned stars six magnitudes numbered from 1 to 6

Range of luminosities
The brightest stars had a magnitude of 1 and the dimmest had a magnitude of 6

Range of luminosities
The magnitudes in between differed from each other by about a factor of two

Range of luminosities
The modern magnitude scale is based on a magnitude 1 star being 100 times brighter than a magnitude 6 star, so magnitudes differ in brightness by about a factor of
2.5

m

apparent magnitude M

absolute magnitude
"absolute magnitude" is the apparent magnitude if the star were at a distance of 10 parsecs (32.6 ly), so it is a measure of the luminosity of the star

Our goals for learning:
• How do we measure stellar luminosities?
• How do we measure stellar temperatures?
• How do we measure stellar masses?

Every object emits thermal radiation with a spectrum that depends on its temperature.

• An object of fixed size grows more luminous as its temperature rises.
Relationship Between Temperature and Luminosity

• An object of fixed size grows more luminous as its temperature rises.
• The color of the light also changes
Relationship Between Temperature and Luminosity

1.
Hotter objects emit more light per unit area at all frequencies.
2.
Hotter objects emit photons with a higher average energy.

The surface temperature of an object can be determined from the peak of its thermal radiation curve using Wien’s Law:
T (K)

λ

(nm)
6

Hottest stars:
50,000 K
Coolest stars:
3,000 K
(Sun’s surface is 5,800 K)

10 6 K
10 5 K
10 4 K
10 3 K
10 2 K
10 K
Ionized
Gas
(Plasma)
Level of ionization also reveals a star’s temperature.
Neutral Gas
Molecules
Solid

Absorption lines in a star’s spectrum tell us its ionization level.

Lines in a star’s spectrum correspond to a spectral type that reveals its temperature:
(Hottest) O B A F G K M (Coolest)

(Hottest) O B A F G K M (Coolest)
• Oh, Be A Fine Girl/Guy, Kiss Me

Thought Question
Which of the stars below is hottest?
A. M star
B. F star
C. A star
D. K star

Thought Question
Which of the stars below is hottest?
A. M star
B. F star
C. A star
D. K star

• Annie Jump
Cannon and the
“calculators” at
Harvard laid the foundation of modern stellar classification.

Orbit of a binary star system depends on strength of gravity

Types of Binary Star Systems
• Visual binary
• Eclipsing binary
• Spectroscopic binary
About half of all stars are in binary systems.

We can directly observe the orbital motions of these stars.

We can measure periodic eclipses.
Exploring the Light Curve of an Eclipsing Binary Star System

We determine the orbit by measuring Doppler shifts.

Isaac Newton
We measure mass using gravity.
Direct mass measurements are possible only for stars in binary star systems.
4π 2 p 2 = a 3
G ( M
1
+ M
2
) p = period a = average separation

Need both p and a to determine mass
1. Orbital period ( p )
2. Orbital separation ( a or r = radius) v
3. Orbital velocity ( v ) r
For circular orbits, v = 2 p r / p
M

Most massive stars:
100 M
Sun
Least massive stars:
0.08 M
Sun
( M
Sun is the mass of the
Sun.)

• How do we measure stellar luminosities?
— If we measure a star’s apparent brightness and distance, we can compute its luminosity with the inverse square law for light.
— Parallax tells us distances to the nearest stars.
• How do we measure stellar temperatures?
— A star’s color and spectral type both reflect its temperature.

• How do we measure stellar masses?
— Newton’s version of Kepler’s third law tells us the total mass of a binary system, if we can measure the orbital period ( p ) and average orbital separation of the system
( a ).

Our goals for learning:
• What is a Hertzsprung–Russell diagram?
• What is the significance of the main sequence?
• What are giants, supergiants, and white dwarfs?

Temperature
An H-R diagram plots the luminosities and temperatures of stars.

Generating an H-R Diagram

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

Large radius
•Stars with lower T than main-sequence stars with the same
L , or with higher L than main-sequence stars with the same
T , must have larger radii

Large radius
•Stars with lower T than main-sequence stars with the same
L , or with higher L than main-sequence stars with the same
T , must have larger radii
•This is because more of the cooler surface is required to give the same luminosity as a hotter star

Large radius
•Stars with lower T than main-sequence stars with the same
L , or with higher L than main-sequence stars with the same
T , must have larger radii
•This is because more of the cooler surface is required to give the same luminosity as a hotter star
•These are giants and supergiants

Small radius
•Stars with higher T than main-sequence stars with the same
L , or with lower L than main-sequence stars with the same
T , must have smaller radii

Small radius
•Stars with higher T than main-sequence stars with the same
L , or with lower L than main-sequence stars with the same
T , must have smaller radii
•These are white dwarfs

A star’s full classification includes spectral type and luminosity class

A star’s full classification includes spectral type (based on spectral line identities) and luminosity class

Lines in a star’s spectrum correspond to a spectral type that reveals its temperature:
(Hottest) O B A F G K M (Coolest)

A star’s full classification includes spectral type (based on spectral line identities) and luminosity class ( based on spectral line shapes, which are related to the size of the star )

“Pressure Broadening” is the basis of luminosity classification increasing size, decreasing pressure
Source: http://spiff.rit.edu/classes/phys440/lectures/lumclass/lumclass.html

“Pressure Broadening” is the basis of luminosity classification
Source: http://spiff.rit.edu/classes/phys440/lectures/lumclass/lumclass.html
increasing size, decreasing pressure
Why do you think increasing the size would decrease the pressure at the surface of the star?

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

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
D
C
Which star is a main-sequence star?
Temperature

A
B
D
C
Which star is a main-sequence star?
D
Temperature

A
B
D
C
Which star has the largest radius?
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 mainsequence star determines its luminosity and spectral type!

Low-mass stars
High-mass stars
The mass of a normal, hydrogenburning mainsequence star determines its luminosity and spectral type!
The reason is gravitational equilibrium, also called hydrostatic pressure

Hydrostatic Equilibrium
The core pressure and temperature of a higher-mass star need to be higher in order to balance gravity.
A higher core temperature boosts the fusion rate, leading to greater luminosity.

Stellar Properties Review
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

Stellar Properties Review
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

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years
Life expectancy of a 10 M
Sun star:

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years
Life expectancy of a 10 M
Sun star:
10 times as much fuel, uses it 10 4 times as fast:

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years
Life expectancy of a 10 M
Sun star:
10 times as much fuel, uses it 10 4 times as fast:
~10 million years

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years
Life expectancy of a 10 M
Sun star:
10 times as much fuel, uses it 10 4 times as fast:
~10 million years
Life expectancy of a 0.1 M
Sun star:

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years
Life expectancy of a 10 M
Sun star:
10 times as much fuel, uses it 10 4 times as fast:
~10 million years
Life expectancy of a 0.1 M
Sun star:
0.1 times as much fuel, uses it 0.01 times as fast:

Mass and Lifetime
Sun’s life expectancy until core hydrogen (10% of total) is used up:
~10 billion years
Life expectancy of a 10 M
Sun star:
10 times as much fuel, uses it 10 4 times as fast:
~10 million years
Life expectancy of a 0.1 M
Sun star:
0.1 times as much fuel, uses it 0.01 times as fast:
~100 billion years

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.

• 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.

• 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.
• Most stars end up small and white after fusion has ceased: white dwarfs.

Relationship between Main-Sequence Stellar Masses and
Location on H-R Diagram

Main-sequence stars (to scale) Giants, supergiants, 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
D
Which of these stars will have changed the least
10 billion years from now?
C

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

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

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

• What is a Hertzsprung–Russell diagram?
— An H-R diagram plots the 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.

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

Globular cluster: Up to a million or more stars in a dense ball bound together by gravity

Visual Representation of a Star Cluster Evolving
Massive blue stars die first, followed by white, yellow, orange, and red stars.

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

The mainsequence turnoff point of a cluster tells us its age.

To determine accurate ages, we compare models of stellar evolution to the cluster data.
Using the H-R Diagram to Determine the Age of a Star Cluster

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.