Chapter 10
Measuring the Stars
Read material in Chapter 10
Some of the topics included in this chapter
• Stellar parallax
• Distance to the stars
• Stellar motion
• Luminosity and apparent brightness of stars
• The magnitude scale
• Stellar temperatures
• Stellar spectra
• Spectral classification
• Stellar sizes
• The Herzsprung-Russell (HR)diagram
• The main sequence
• Spectroscopic parallax
• Extending the cosmic distance
• Luminosity class
• Stellar masses
Parallax is the apparent shift of an object relative to some distant
background as the observer’s point of view changes
It is the only direct way to measure distances to stars
It makes use Earth’s orbit as baseline
Parallactic angle = 1/2 angular shift
A new unit of distance: Parsec
By definition, parsec (pc) is the distance from the Sun
to a star that has a parallax of 1” (1 arc second)
Parallax Formula:
Distance (in pc) = 1/parallax (in arcsec)
One parsec = 206,265 AU or ~3.3 light-years
As the distance increases to a star, the parallax decreases….
Examples using the parallax formula:
If the measured parallax is 1 arcsec, then the distance of the star is 1 pc. (Distance 1 pc = 1/1 arcsec)
If the measured parallax is 0.5 arcsec, then the distance of the star is 2 pc. (Distance 2 pc = 1/0.5 arcsec) Note: 1 parsec = 3.26 light-years.
The Solar Neighborhood
Let’s get to know
our neighborhood:
A plot of the 30
closest stars within
4 parsecs (~ 13 ly)
from the Sun.
The gridlines are
distances in the
galactic plane (the
plane of the disc of
the Milky Way)
More examples using the parallax formula
The nearest star Proxima Centauri (the faintest star of the triple star
system Alpha Centauri) has a parallax of 0.76 arcsec.
• Therefore, distance = 1 / 0.76 = 1.32 pc (4.29 ly)
• The next nearest star is Barnard’s star, with a parallax of 0.55”
• Therefore, d = 1 / 0.55 = 1.82 pc (5.93 ly)
The Nearest Stars
• From the ground, we can measure parallactic angles of ~1/30 (0.03”) arcsec, corresponding to distances out to ~30 pc (96 ly).
• There are several thousand stars within that distance from the Sun.
• From space (Hipparcos satellite), parallax’s can be measured down to
about 5/1000 arcsec, which corresponds to 200 pc (~660 ly).
• There are several million stars within that distance.
Using the stellar parallax, the distance to these stars can be determined
directly
Stellar Proper Motion
• Parallax is an apparent motion of stars due to Earth
orbiting the Sun.
• But stars do have real space motions.
• Space motion has two components:
1) “line-of-sight” or radial motion (measured through Doppler
shift of emission/absorption lines)
2) “transverse” motion (perpendicular to the line of sight)
observer star
radial motion
transverse space motion
How to determine the two component of the
space motion of a star?
• Use the Doppler shift to determine the radial component. Observe the
shift in wavelengths of the emission or absorption lines. Then apply
the formula of Doppler shift to determine the radial velocity
• Use the proper motion and the distance to determine the transverse
component.
First we need to measure the proper motion. Proper motion is
measured in arc seconds/year. Then we need to know the distance to
the star using parallax so we can determine the transverse component
• Finally use trigonometry to calculate the transverse velocity
• This method works for stars that are nearby so we can measure the
proper motion.
• The total velocity can be calculated using the Pythagorean theorem:
____________________________________
Total velocity = √ [(Radial velocity)² + (Transverse velocity)²]
Stellar Proper Motion: Barnard’s Star
• Two pictures, taken 22 years apart ( Taken at the same time of the year
so it doesn’t show parallax!). Barnard’s star is a red dwarf of magnitude +9.5,
invisible to the naked eye (limit of naked eye is +6)
• Barnard’s star has a proper motion of 10.3 arcsec/year (it is the
star with the largest proper motion)
• Given d = 1.8 pc, this proper motion corresponds to a
“transverse” velocity of ~90 km/s !
Question: What does the proper motion depend on? Answer 1: Space velocity
Answer 2: Distance
Some important definitions and
concepts
• Luminosity is the amount of radiation leaving a star per unit time.
• Luminosity is an intrinsic property of a star.
• It is also referred as the star absolute brightness.
It doesn’t depend on the distance or motion of the observer respect to
the star.
• Apparent brightness or Flux.
When we observe a star we see its apparent brightness, not its
luminosity. The apparent brightness (or flux) is the amount of light
striking the unit area of some light sensitive device such as the human
eye or a CCD. It depends on the distance to the star.
Apparent Brightness and the Inverse Square Law: Proportional to 1/d2
• Light “spreads out” like the
distance squared.
• Through a sphere twice as large,
the light energy is spread out over
four times the area.
(area of sphere = 4d2)
The apparent brightness or Flux
decreases with distance, it is
inversely proportional to the
square of the distance.
It can be determined by:
Luminosity
4d2 Flux =
To know a star’s luminosity we must measure its apparent
brightness (or flux) and know its distance. Then,
Luminosity = Flux *4d2
Luminosity and Apparent Brightness
Two stars A and B of
different luminosity
can appear equality
bright to an observer if
the brightest star B is
more distant than the
fainter star A
The Magnitude Scale 2nd century BC, Hipparchus ranked all visible
stars
He assigned to the brightest star a magnitude
1, and to the faintest a magnitude 6.
Later, astronomer found out that a difference
of 5 magnitudes from 1 (brightest) to 6
(faintest) correspond to a change in brightness
of 100
To our eyes, a change of one magnitude = a
factor of 2.512 in flux or brightness.
The magnitude scale is logarithmic.
Each magnitude corresponds to a factor of
1001/5 2.5
5 magnitudes = factor 100 in brightness.
The apparent magnitude scale was later
extended to negative values for brighter
objects and to larger positive values for
fainter objects
Brightest
Faintest
Equivalence between magnitude and brightness Magnitude Brightness
-1
2.512
0
2.512
1
2.512
2
2.512
3
2.512
4
2.512
5
2.512
6
The change of brightness between magnitude 1 and 6 is 2.512^5 = 100
In general, the difference in brightness between two magnitudes is:
Difference in brightness = 2.512 ^n, where n is the difference in magnitude
Example: What is the difference in brightness between magnitude -1 and +1?
Answer: n=2, difference in brightness = 2.512² = 2.512 x 2.512 = 6.31
Absolute Magnitude is the apparent magnitude of a star as
measured from a distance of 10 pc (33 ly).
Sun’s absolute
magnitude = +4.8
It is the magnitude of
the Sun if it is placed at
a distance of 10 pc.
Just slightly brighter
than the faintest stars
visible to the naked eye
(magnitude = +6) in the
sky.
Enhanced color picture of the sky
Notice the color differences among the stars
Stellar Temperature: Spectra
• The spectra shows 7 stars with
same chemical composition but
different temperatures.
• Different spectra result from
different temperatures.
Example: Hydrogen absorption lines
are relatively weak in the hottest star
because it is mostly ionized.
Conversely, hotter temperatures are
needed to excite and ionize Helium so
these lines are strongest in the hottest
star.
Molecular absorption lines (TiO) are
present in low temperature stars. The
low temperatures allow formation of
molecules Ti Titanium, TiO titanium oxide
Spectral Classification:
Annie Jump Cannon
The stars were classified by the Hydrogen line strength,
and started as A, B, C, D, …
But after a while they realized that there is a sequence in
temperature so they rearranged the letters (some letters were
drop from the classification) so that it reflect a sequence in
temperature. It became:
O, B, A, F, G, K, M, (L)
A temperature sequence! Cannon’s spectral
classification system was officially adopted in 1910.
A classification of
stars was started by
the “Pickering’s
women”, a group of
women hired by the
director of the Harvard
College observatory,
including Annie
Cannon
Spectral Classification
A mnemonic to remember the correct order:
“Oh Be A Fine Girl/Guy Kiss Me”
Each letter is divided in 10 smaller subdivisions from 0 to 9. The lower the number, the hotter the
star. Example, G0 (hotter) to G9 (cooler). The Sun is classified as a G2 star, the surface
temperature is 5800 K
Strengths of Lines at Each Spectral Type
Stellar Radii
• Almost all stars are so distant that the image of their discs look so
small. Their images appear only as an unresolved point of light
even in the largest telescopes. Actually the image shows the Airy
disk produced by the star. • A small number of stars are big, bright and close enough to
determine their sizes directly through geometry.
•Knowing the angular diameter and the distance to the star, it is
possible to use geometry to calculate its size.
Diameter/2π x distance = Angular diameter/360
Stellar Radii • One example in which it is possible to use geometry to determine the radius
is the star Betelgeuse in the Orion constellation
• The star is a red giant located about 640 ly from Earth
• Betelgeuse size is about 600 time larger than the Sun
• Its photosphere exceed the size of the orbit of Mars
• Using the Hubble telescope it is possible to resolve its atmosphere and
measure its diameter directly
• The measured angular size is about 0.043-0.056 arc seconds
An indirect way to determine the stars radii
• Most of the stars are too distant or too small to allow the
direct determination of their size.
• But we can use the radiation laws to make an indirect
determination of their size.
• According to Stefan law, the luminosity of a star is
proportional to the fourth power of the surface temperature
(T4 )
• The luminosity also depend on its surface area. Larger
bodies at the same temperature radiate more energy.
• Luminosity Surface area * T4
Lstar= (Rstar/Rsun)2 * (Tstar/Tsun)
4 * Lsun
Stellar Radii: An indirect way to measure the radius (Read 10-2 More Precisely, “Estimating Stellar Radii’)
Stefan’s Law F = T4
Luminosity (L) is the Flux (F) multiplied
by the entire spherical surface (A)
L = A * F
Area of sphere A = 4R2
Expressing in solar units (dividing
by the solar L, R and T), the
constants disappear:
L = 4R2T4
L = 4R2 F
Flux (F) is the energy radiated per unit area
by a black body at the temperature T
(R is the radius of the star)
Substituting A in the equation of L
Substituting F in the equation of L
The relationship between Luminosity, Radius, and Temperature
provides a means to evaluate these properties relative to the Solar
values.
T
T
R
R
L
TRTR
L
sunsunsun
sunsunsun
L
L
42
42
42
4
4
For example, a star has 10 times the Sun’s radius but is half as hot. (Since this is relative to the Sun, we will consider that the radius of the Sun is 1
and the temperature of the Sun is 1)
How much is the luminosity respect to the Sun?
25.616
100
2
1
1
1042
Lsun
L
Determining radii using radiation laws
The equation L = 4R2T4 can be expressed in solar units as:
• L(in solar luminosities) = R2 (in solar radius) * T4 (in solar surface
temperature)
• If we need to calculate the radius, we can rearrange the equation :
R2 (in solar radius) = L(in solar luminosities) / T4 (in solar surface
temperature)
Here we need to know the luminosity L and T. To determine L, we need
to know the Flux and the distance d. To get T, we need to get the
spectrum of the star.
Luminosity = Flux *4d2
Understanding Stefan’s Law: Radius dependence
Lstar= (Rstar/Rsun)2 * (Tstar/Tsun)
4 * Lsun
If we receive 100 photons from the Sun, we should receive 400 photons from a star twice the
diameter of the Sun. The star will look four times brighter than the Sun
Let’s consider a star that has a radius twice the radius of the Sun.
What will be the luminosity of that star? (We assume that the two stars have the same temperature)
Understanding Stefan’s Law: Temperature dependence
Lstar= (Rstar/Rsun)2 * (Tstar/Tsun)
4 * Lsun
The luminosity of a star that has a temperature twice that of the Sun, must be 16 times
larger.
The luminosity of a star with a temperature 1/3 of the Sun, must be 1/81 that of the Sun
The assumption here is that these stars have the same radius
Let’s consider a star with a temperature twice that of the Sun and
another star with a temperature one third of the Sun
Hertzsprung-Russell (HR) Diagram
The Main Sequence (MS)is the
diagonal band of stars in the HR
diagram
Stars reside in the main sequence
during the period in which the core
burns H
Most stars (like the Sun) lie on the
main sequence. The Sun will spend
most of its life in the main sequence (It
has been in the MS for about 5 billion
years)
The HR diagram is a plot of
star Luminosity versus
Temperature (or spectral
class)
It also give information about:
•Radius
•Mass
•Lifetime
•Stage of Evolution
Main sequence
From Stefan’s law…...
Let’s use the equation and
the HR diagram to learn
more about L, R and T
More luminous stars at
the same T must be bigger!
Cooler stars at the same L
must be bigger!
L = 4R2 T4
The HR diagram to the
right has L and T on
the axes. But we can
plot R (The other
parameter in the
equation) also which
will appear as straight
lines crossing the
diagram
The HR Diagram: 100 Brightest Stars
• Most luminous stars,
because they are so rare, lie
beyond 5 pc.
• If we know the luminosity,
we can determine distance
from their Flux (brightness).
Luminosity
4d2 Flux =
The technique to
determine distances to
stars using the radiation
laws and HR diagram is
called: Spectroscopic
“Parallax”
The HR Diagram: Spectroscopic “Parallax”
Main Sequence
1) We measure the Flux or apparent
brightness of a star
Apparent brightness is the rate at which
energy from the star reaches a detector
2) From the spectrum of a star, we
can determine its temperature or the
spectral type.
3) Then using the HR diagram we can
determine its luminosity assuming
it is located in the Main
Sequence
4) Use inverse square law to
determine distance.
An example to illustrate how this
works:
Luminosity
4d2 Flux =
What if the star doesn’t happen to lie on the Main
Sequence - maybe it is a red giant or white dwarf???
We determine the star’s Luminosity Class based on its
spectral line widths:
Spectral lines
get broader
when the
stellar gas is at
higher
densities -
indicates
smaller star.
A Supergiant
star
A Giant star
A Dwarf star (Main Sequence)
Wavelength
The HR Diagram: Luminosity & Spectroscopic Parallax
The HR Diagram: Luminosity Class
Bright Supergiants
Supergiants
Bright Giants
Giants
Sub-giants
Main-Sequence (Dwarfs)
• Isn’t this getting a little circular?
• First we said that we derive Luminosities from measured Fluxes and
Distances?
• Now we’re saying we know the Luminosities and we use them together with
Temperatures to derive Distances……..
Let’s clarify this!
Example of absorption lines for different spectral classes
The lines are wider for dwarf (denser) stars of spectral class V and
narrower for giant stars of spectral class I.
More on Spectroscopic Parallax
The answer:
• Now we made use of additional information obtained from
the spectral analysis.
• The spectral analysis provide information to determine the
temperature of the star or the spectral classification (Using
the spectrum of the star). To do this, we didn’t know or
need the distance
• Next we also made use of the HR diagram. If we know the
temperature for a main sequence star (or the luminosity
class), then we can deduce the luminosity
We get distances to nearby planets from radar
ranging
If we know the distance (and we can measure
the orbital period), we apply Kepler’s 3rd law to
obtain the distance Earth-Sun (AU)
That sets the scale for the whole solar system
(1 AU). It allows us to get a value for the AU in
km (1AU = 150,000,000 km)
Knowing the value of the AU in km, we use
the stellar parallax, to find distances to “nearby”
stars.
Use these nearby stars with known distances, then we measure the Fluxes and determine the
Luminosities, to calibrate Luminosity classes in HlR diagram. In other words, one uses nearby stars
for which one can determine the stellar parallax and also the spectroscopic parallax .
Then for farther stars, knowing spectral class (or T) one can determine Luminosity. Next one
measure the Flux and get Distances (Spectroscopic Parallax).
The spectroscopic parallax is useful to determine distances within our galaxy
The Distance Ladder
With Newton’s modifications to Kepler’s laws, the period and size of the orbits yield
the sum of the masses.
P² = a³ /(m1 + m2 )
The relative distance of each star from the center of mass yields the ratio of the
masses.
m1d1 = m2d2
The ratio and the sum of the masses provide the information to calculate each mass
individually (Two equations and two unknowns). P, a, d1 and d2 are known (these four
parameters can be measured)
Note: For Sirius, the plane of the orbit is not face on, it is inclined 46 degrees from the line of sight. A
correction needs to be done first before using the values of size of orbit and distance to the center of mass
Stellar Masses:
Visual Binary Stars Binary star are classified as
visual, spectroscopic and
eclipsing The example shows Sirius (visual
binary), the brightest star in the sky.
Sirius A has a companion Sirius B, a
very dense object called white dwarf
Stellar Masses: Spectroscopic Binary Stars
In this example, using a telescope the observer cannot resolve the two stars and see the two stars as a single
star…
An example: The multiple star Castor. In a telescope one can see two stars. Each one of the two is a
spectroscopic binary. There is a third, fainter star in the Castor system which is also a spectroscopic binary
Many binaries are too far away or they orbit around the other star at a short
distance, but they can be discovered from periodic spectral line shifts. The shift in
wavelenght of the spectral lines as they orbit each other show a Doppler effect
Stellar Masses: Eclipsing Binary Stars
How do we identify eclipsing binaries?
We can identify an eclipsing binary by observing the light curve of the star, a plot of the apparent
brightness of the star as function of time
The occultation of the star in the system must be observed only if we can see the orbital plane
“edge on”.
This method also tells us something about the stellar radii (Through the deep of the eclipse).
The HR Diagram: Stellar Masses
Why is the mass of a star so important?
Together with the initial composition, mass defines the entire life
cycle and all other properties of the star!
The mass of a star will determine:
• Luminosity
• Radius
• Surface Temperature
• Lifetime
• Evolutionary phases
• And how the star will end its life….
All of this is determined by the mass of the
star. A note: The composition of the first stars was H and He. Heavier elements are
produced in the interior of the stars. After the interstellar gas was contaminated with
heavier elements produced in the interior of the first stars (Example: Supernova), the
composition of the mass of the later generation of stars incorporated those heavier
elements
Example: For stars on the Main Sequence, if
we plot the luminosity as a
function of mass, we find that the
luminosity depends of the mass (Notice that this plot is in log scale )
Luminosity Mass4
Why the luminosity increases
at such high rate?
A star with more mass means:
• more gravity
• more pressure in the core
• higher core temperatures
• faster nuclear reaction rates
• fast production of energy ( Mass4)
• higher luminosities!
• shorter lifetime
Lifetime Fuel available / How fast fuel is burned
Stellar lifetime Mass / Mass4 = 1 / Mass3
Or, since Luminosity Mass4
Stellar lifetime Mass / Luminosity
So for a star
(For main sequence stars)
How long a star lives is directly related to the mass!
Example: The Sun lifetime is estimated to be about 10 billion years.
A star with 10 times the mass of the Sun has an estimated lifetime of
10 million years!
Do the calculation!
Big (Massive) stars live shorter lives, burn their fuel
faster….
H-R diagram Location of stars of different masses
Stars of large mass will evolve fast and move off the main sequence faster that low mass stars
The turn off point of two open star clusters in
the H-R diagram showing their different ages
The turn off points: Points where the stars are moving
off the main sequence
Stars of higher mass leave the
Main Sequence earlier
What can we deduce from the
HR diagram and the turn off
points about the relative age of
these two clusters?
Cluster M 67 is younger than
NGC 188. Massive stars in M 67
are still in the main sequence.
Stars of similar mass in NGC
188 are off the main sequence