# Chapter 15 part 1

```Chapter 14
Surveying the Stars
Luminosity and Apparent Brightness
Luminosity - the measurement of the rate at which energy is outputted from a
star. This could also be called the power output of the sun.
=  4 = 4 2  4
Where:
= 5.67•10−8

2  4
(Stefan-Boltzmann constant)
= 4 2 = The surface area of the star with a radius
= the temperature of the star in Kelvin
However, when we look at a star, we see not its luminosity but rather its
apparent brightness
Apparent brightness is a measure not of a star’s luminosity but of the energy
flux (energy per unit area per unit time) produced by the star, as seen from
Earth. It depends on our distance from the star.
Thought Question
These two stars have about the same luminosity -which one appears brighter?
A. Alpha Centauri
B. The Sun
Luminosity:
Amount of power a star radiates
(energy per second = Watts)
Apparent brightness:
Amount of starlight that reaches
Earth
(energy per second per square
meter)
Luminosity passing through each
sphere is the same
Divide luminosity by the area of
the sphere over which the light is
being spread to get the apparent
brightness at every point on that
sphere.
The further from the star your go
the larger the area over which the
light is spread so the dimmer the
star will appear to be.
The relationship between apparent brightness and luminosity depends on
distance:
Brightness =
Luminosity
4π (distance)2
Thus we can determine a star’s luminosity if we can measure its distance and
apparent brightness:
Luminosity = 4π (distance)2 x (Brightness)
Thought Question
How would the apparent brightness of Alpha
Centauri change if it were three times
farther away?
A.
B.
C.
D.
It would be only 1/3 as bright
It would be only 1/6 as bright
It would be only 1/9 as bright
It would be three times brighter
Most luminous stars:
106 LSun
Least luminous stars:
10-4 LSun
(LSun is luminosity of Sun)
So how far are these stars?
Parallax
Apparent
positions of
nearest stars
an arcsecond as
Earth orbits
Sun
Parallax angle
depends on
distance
Parallax and Distance
p = parallax angle
d (in parsecs)
d (in light
- years)
=
1
p (in arcseconds)
= 3.26 
1
p (in arcseconds)
Our nearest neighbours
The closest star to Earth (besides the Sun) is called Proxima Centauri. It has the largest
known stellar parallax, 0.76'', which means that it is about 1/0.76 = 1.3 pc away—
about 270,000 A.U., or 4.3 light-years.
Our next nearest neighbor to the Sun beyond the Alpha Centauri system is called
Barnard’s Star. Its parallax is 0.55'', so it lies at a distance of 1.8 pc, or 6.0 light-years.
Stellar Motion we must be careful
with parallax.
Barnard’s Star 22 years apart:
This stellar motion has two components. A star’s radial velocity—along the line of
sight—can be measured using the Doppler effect. For many nearby stars, the
transverse velocity—perpendicular to our line of sight—can also be determined by
careful monitoring of the star’s position on the sky.
The annual movement of a star across the
sky, as seen from Earth and corrected for
parallax, is called proper motion. It describes
the transverse component of a star’s velocity
relative to the Sun. Like parallax, proper
motion is measured in terms of angular
displacement
Spectral lines from Alpha Centauri are
blueshifted by a tiny amount—about 0.0067
percent—allowing astronomers to find the
radial velocity from the Doppler Effect
Use basic trig to find the
hypotenuse.
THE MAGNITUDE SCALE
The scale dates from the second century B.C., when the Greek astronomer
Hipparchus ranked the naked-eye stars into six groups. The brightest stars
were categorized as first magnitude.
The 1–6 magnitude range defined by Hipparchus spans about a factor of 100
in apparent brightness—a first-magnitude star was approximately 100 times
brighter than a sixth-magnitude star.
The physiological characteristics of the human eye are such that each
magnitude change of 1 corresponds to a factor of about 2.5 in apparent
brightness. [ 2.512 to the fifth power is 100]
We now define a change of 5 in the magnitude of an object (going from
magnitude 1 to magnitude 6, say, or from magnitude 7 to magnitude 2) to
correspond to exactly a factor of 100 in apparent brightness.
Because we are really talking about apparent (rather than absolute) brightnesses,
the numbers in Hipparchus’s ranking system are called apparent magnitudes.
The scale is no longer limited to whole numbers
Absolute magnitude
To compare intrinsic, or absolute, properties of stars, however, astronomers
imagine looking at all stars from a standard distance of 10 pc (arbitrary choice).
Because the distance is fixed in this definition, absolute magnitude is a
measure of a star’s absolute brightness, or luminosity.
Knowing distance allows us to compute its absolute magnitude. As discussed
further in lab the numerical difference between a star’s absolute and apparent
magnitudes is a measure of the distance to the star
The Magnitude Scale
m = apparent m agnitude
M = absolute m agnitude
apparent brightness of S tar 1
 (100
1/ 5
)
m1  m 2
 2.51
m1  m 2
apparent brightness of S tar 2
lum inosity of S tar 1
lum inosity of S tar 2
 (100
1/ 5
)
M1M 2
 2.51
M1M 2
How do we measure stellar
temperatures?
1. Hotter objects emit more light per unit area at all
frequencies. (Brighter)
2. Hotter objects emit photons with a higher average energy.
(Bluer)
Hottest stars:
50,000 K
Coolest stars:
3,000 K
(Sun’s surface
is 5,800 K)
Absorption lines in star’s spectrum tell us ionization level
which is highly dependent on temperature.
Pioneers of Stellar Classification
• Annie Jump
Cannon and the
“calculators” at
Harvard laid the
foundation of
modern stellar
classification
Early computers were not allowed
to be Laptops
Harvard “computers”
were ladies who did the
tedious data analysis by
hand.
1897: Antonia Maury
spectral study led to HR
diagrams.
1898: Annie Cannon’s
spectral classes still
used. She ranked stellar
spectra based on the
strength of the
Hydrogen spectral lines.
1908: H. Leavitt’s
variable star work .
Lines in a star’s spectrum correspond to a spectral type that reveals its temperature
The original classification of spectral type was alphabetical and based on the strength of
the hydrogen lines. Spectra with the strongest hydrogen lines were call A type stars.
Spectral Types in Temperature Order
(Hottest)
O B A F G K M
(Coolest)
•Only Boys Accepting Feminism Get Kissed
Meaningfully
Each spectral type is subdivided into numbered subcategories (0-9)
0 being the hottest and 9 the coolest.
For example; if we break the F spectral type stars into its
subcategories the order would look like
O B A (F0, F1, F2…, F9) G K M
The sun is a G2 spectral class star.
How do we measure stellar
masses?
The 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
Visual Binary
We can directly observe the orbital motions of these
stars
Eclipsing Binary
We can measure periodic eclipses
Spectroscopic Binary
We determine the orbit by measuring Doppler shifts
Alcor and Mizar are optical doubles not true binaries. They just look
close. But Mizar A & B is a true visual binary made up of two
spectroscopic binaries (4 stars total).
Estimating Stellar Masses
Recall Kepler’s 3rd Law:
4 2 3
2
=
(1 + 2 )
If we choose to measure the Period in
years, the average orbital separation in
astronomical units (AU), and the
masses in solar masses then the
constants
42

=1
So
3
1 + 2 = 2

Example:
A binary system is observed with a period of P = 32 years
and separation a of a = 16 AU:
3
1 + 2 = 2

163
1 + 2 = 322 = 4 solar masses
Need 2 out of 3 observables to
measure mass:
1) Orbital Period (P)
2) Orbital Separation (a or r = radius)
3) Orbital Velocity (v)
v
For circular orbits, v = 2pr / p
r
M
Most massive
stars:
100 MSun
Least massive
stars:
0.08 MSun
(MSun is the mass
of the Sun)
Back to Stellar Masses.
• We can add mass to the data we have on stars
thanks to the binaries.
• We find…
• Supergiants, giants and white dwarfs reveal no
clear pattern.
• But main sequence stars show a clear trend.
MAIN SEQUENCE STARS ONLY
(a) Size and mass relation
(b) Luminosity and mass
What have we learned?
• 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
What have we learned?
• 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)
```