# week12

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```					                 Flux vs. Luminosity
• luminosity (L): rate of energy release
If L is constant, equal amounts of energy will flow through
each sphere each second…
BUT:
flux (F): rate of energy reaching
each square meter of surface

…same energy is spread over a
larger area

L
flux : F 
4d 2

SURFACE AREA OF A SPHERE
Thought Question:
Imagine you are comparing the brightness of two
stars. Star A’s luminosity is 5 times higher than star
B’s, and star A is 3 times farther away from you than
star B. How does the brightness of star A compare
to the brightness of star B? (Try a ratio…)

1.   Star A is 5/3 as bright as star B.
2.   Star A is 5/9 as bright as star B.
3.   Star A is 9/5 as bright as star B.
4.   Star A is 3/5 as bright as star B.

L
F
4d 2
Brightness
Earth’s orbit

Sun
Thought Question:
If you were standing on the dwarf planet Eris (67.7 AU from
the Sun on average), how would the Sun’s brightness
compare to what we measure on Earth?
Planet Temperatures
Planets are in thermal equilibrium:           SUNLIGHT

HEATING BY = COOLING BY
SUNLIGHT     PLANET LIGHT

Both Sun and planets emit

PLANET LIGHT
(INFRARED)
Planet Temperatures
Planets are in thermal
equilibrium:

HEATING BY
SUNLIGHT   = COOLING BY
PLANET LIGHT

Both Sun and planets emit
Thought Question:
How would the equilibrium temperature of Earth change if the

SOURCE OF HEAT
(SUNLIGHT)                           HEAT LOST     (PLANET

HEAT NOW IN
PLANET

L(1 a)
T4
16d 2
Planet Temperatures
SUNLIGHT

PLANET LIGHT
(INFRARED)
Thought Question:
How would the equilibrium temperature on the dwarf planet
Eris (67.7 AU from the Sun on average) compare to the
temperature on Earth on Earth if they reflect the same
percentage of sunlight?
L(1 a)
T4
16d 2


Equilibrium Temperature of the Planets
• distant planets are cold mainly because of
L
F
inverse square law for light:

4d 2


Luminosity (L)
maximum        Sun       minimum
106 L       1 L       10-4 L

centi-firefly?

How to calculate L for stars:
• measure brightness at Earth
• measure distance                  L
• use inverse-square law:       F
4d 2
O    Surface Temperature
B             • star colors change as
temperature changes
A
Compared to Sun:
F              hotter stars look
blue-white
G (Sun)        cooler stars look red

K              Sun is actually white
color
M
Surface Temperature (T)
maximum       Sun       minimum
105 K     5800 K       2800 K

How to measure:
• overall color or most intense wavelength

(2.9 10 6 nm  K )
peak 
T

• spectral lines
Spectral Types
hottest
• pattern of
absorption lines
reveals star
temperature

barcode or a
fingerprint

coolest
Sirius

… has a low-powered star
orbiting it:

SIRIUS B:
• 50 yr orbit period
• temperature: 25000 K!
• 1/40th Sun’s luminosity!

SIRIUS B (Hubble Space Telescope image)
Sun:
radius: R = 7  105 km
measured by knowing
distance from Earth and its
angular size

0.5º

1 AU

.
Jupiter   Earth
0.1 R    0.01 R
Temperature, Size, Luminosity
Two things can increase LUMINOSITY of a star:

EQUAL AREA PATCHES:           EQUAL TEMPERATURE PATCHES:
HOTTER ONE RELEASES           LARGER AREA RELEASES
MORE LIGHT PER SECOND         MORE LIGHT PER SECOND

HOT            COOL
Temperature, Size, Luminosity
Two things can increase LUMINOSITY of a star:

A HOTTER OBJECT RELEASES      A LARGER AREA RELEASES MORE
MORE LIGHT PER SECOND         LIGHT PER SECOND:
FROM EACH BIT OF SURFACE

HOT            COOL

SAME AREA                SAME TEMPERATURE
Star Sizes
 brightness of each piece of surface
only depends on temperature

L
flux from star’s
 F  T 4

4R 2
surface
flux from thermal

L  4R T
Boltzmann Law)

We can calculate the size of the star!

Thought Question:
The stars Antares and Mimosa each have the
same luminosity, but Antares is cooler than
Mimosa. Which star is larger?
1. Antares
2. Mimosa
3. They will be the same size.
4. This is not enough information to decide.

L  4R T
2    4
Dwarfs and Giants
If a star is LUMINOUS but COOL:
LUMINOSITY of a star
depends on:
• surface temperature
• size

L  4R 2T 4

If a star is HOT but LOW LUMINOSITY

it must be small (small surface area)
it must be big (large surface area)
 WHITE DWARF
 GIANT
The HR
Diagram

…a “snapshot” of
star properties

Star properties
change very
slowly, so we
can’t see them
change…
Thought Question
In the graph below, which star (represented by dots)
must have the smallest size?
Luminosity
1                2

3

4                5

Temperature
The HR
Diagram
Luminosity

…a “snapshot” of
star properties

Star properties
change very
slowly, so we
can’t see them
change…

Temperature
Thought Question:
A star is about 100 times less luminous than the
Sun, but has a surface temperature that is about
twice as hot as the Sun. How does the star’s
size compare to the Sun? (Hint: write down
equations for both stars and take a ratio…)
Thought Question:
If you took a star that was the same mass as
the Sun and made it 10 times smaller (in
diameter), its density would
1.   remain the same.
5. become 100x larger.
2.   become 10x smaller.
6. become 1000x smaller.
3.   become 10x larger.
7. become 1000x larger.
4.   become 100x smaller.

VOLUME = LENGTH                         4
VOLUME = r 3
WIDTH  HEIGHT                     3


maximum            Sun        minimum
main sequence
stars:
.

20 R      1 R      0.1 R
(Jupiter-size)

How to calculate:
• blackbodies:  brightness of thermal
radiation at star’s surface:   F  T   4

 average brightness released by surface:          L
F
4R 2
So:                          L
R           
4 T 4
Star
Census
Measuring Mass
 Important to measure in
astronomy…
Understanding gravity allows us to:                     v
• distinguish between different types of                    m
planets                                             r
• predict future lives of stars based on mass   M
• quickly measure how much “stuff” is in
something very big (like the Milky Way)
• find matter even when it is “dark”
The Center of the Milky Way

Near galactic center:
•moving stars appear to
be orbiting something
dark…

…almost 4 x 106 Msun!

SIZE OF PLUTO’S ORBIT
Need 2 out of 3 to measure
total mass:
1) orbit period (P)
2) orbit size (a or r = radius)            v

3) orbital speed (v)                           m
r
M
For circular orbits, v = 2r / P
Kepler’s Third Law:
4    2
P 
2
a 3

G ( M  m)
Measuring Mass
Mass inside the orbit can be                                           2
rv
measured using speed and distance                      M
G
v1                                   v2
m                                        m
r                                      r
M1                                   M2

SAME DISTANCE r:
 force exerted by mass M1 must be larger than mass M2

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