Name _______________________________
Partners _____________________________
How far away are the stars and how bright are they, really?
We found previously that the distance d, apparent brightness b, and the luminosity L of a star
are related by equation (4) in Light and Starlight. We can invert that equation to get
d
L
(1)
4 b
So, if we know the luminosity we can find the distance, since b just has to be measured here
on earth. This means that a star that looks dim might be really luminous, but very far away.
And, very close stars will typically look bright because they are close, but may not be that
luminous. So, there is ambiguity in determining d or L. However, if stars are close enough
to the earth there is a direct way to determine the distance to the star – through stellar
parallax.
Parallax
Suppose we look at an object in the room, and move our head side to side. We notice that the
object seems to move relative to the background. If we were to displace that object farther
back from us, then the relative motion is reduced. This is just due to geometry.
1. Your instructor has positioned 3 “stars” in sequence (yellow, orange, green). Aligned with
them on the floor is a center point (orange), and a right point (green) 20cm from the left
point. This represents the radius of earth’s orbit, for example. So, stand at the “center
point”. Note the position on the far wall all the “stars” are aligned on. Then step to the right
and stand at the “right point”. Each star now is displaced from the initial position. Measure
each displacement (in cm) at arm’s length enter in table I.
2. Now, measure the distance from your forehead/eyes to your measuring scale (cm)
(i.e., “arm’s length”) _______________________
Table I – distance computation using parallax
Star
displacement
(cm)
Parallax angle
(eq.2)
d (cm)
(eq.3)
d (cm)
(actual)
d (AU)
10cm =1AU
yellow
orange
green
The parallax angle is “p” in Fig.1. It is computed using trigonometry where the background
is much farther than the star, and can be obtained from
p = tan-1(displacement/arm’s length)
1
(in degrees)
(2)
4. Compute (4) using the data your calculator and enter as the “parallax angle” in table I.
b
a
c
k
g
r
o
u
n
d
“star”
observer
p
y
d
Fig.1 Parallax angle p, where y is 20m in the classroom observation.
We can compute the distance d to the object from Fig.1 also using trigonometry:
d = y/tan(p)
(3)
5. Use Compute d for yellow, orange and green stars in cm using y = 20cm, and also in AU
where 10.0cm = 1AU, and enter into table I.
The measurements of d are known, and will be given to you by the instructor. Note that 30
parallax produces 30AU distance in this model.
Stellar Parallax
As the Earth moves back and forth in its orbit and we view a star, the star seems to move
relative to the “distant stars” behind it. The farther the star is, the less it moves. This
apparent motion of the stars relative to very distant stars is called “stellar parallax” (see
Fig.2). In fact, if p is in seconds, we can compute that
d = 1/p (in pc)
(4)
where p is the parallax angle in seconds, we get the distance in “parsecs” (pc). “1 pc” is
equal to 3.26 ly (about 206,000 AU). It is the distance to a star whose parallax angle is
exactly 1 sec of arc. Remember, a circle is 3600, and 10 is 60 min, each min is 60 sec. This
means that 1sec of arc is 1/3600th of a degree.
The nearest star Proxima Centauri has the largest p of 0.77 sec of arc. Then (4) yields
d = (1/0.77) pc = 1.3pc
= 4.2 ly
Only stars within 100pc give us reliable distances for earth based parallax. This is a lot
of stars, maybe a million, that are within that distance. All are within the near neighborhood
of our sun, of course (remember, the center of the galaxy is about 8000pc away).
2
distant
stars
star
sun
d
p
1-AU
E
Fig.2 As the earth orbits the sun, stars a distance d away have an apparent movement
through an angle 2p, where p is the “parallax angle”. The bigger d is the smaller p is.
6. In Table II are the brightest stars in the sky [Adapted from Norton's 2000.0, 18th edition]. For
each star, enter its constellation, parallax angle p, temperature (for later), and compute d (in pc)
from equation (4).
Table II – the Brightest Stars visible in the northern hemisphere. m is apparent magnitude, p parallax (in milliarc-sec), d distance, M absolute magnitude, T surface temperature, L luminosity and R radius (compared to sun)
Star
Sun
Sirius
Arcturus
Vega
Capella
Rigel
Procyon
Betelgeuse
Altair
Aldebaran
Antares
Spica
Pollux
Deneb
Regulus
Castor
Scientific
name
Constellation
m
-
-
-26.72
-1.46
-0.04
0.03
0.08
0.18
0.38
0.50
0.77
0.85
0.96
0.98
1.14
1.25
1.35
1.97/2.91
αCMa
αBoo
αLyr
αAur
βOri
αCMi
αOri
αAql
αTau
αSco
αVir
βGem
αCyg
αLeo
αGem
p
(msec)
d (pc)
T(K)
M
L
R
-
-
5800
4.83
1
1
HR region
379
90
129
77
4.2
286
5.1
194
50
5.4
12.4
97
2.3
42
66
For stars more distant than 100pc, distances are obtained from satellite observation at much
larger orbits. The satellite known as Hipparcos (high precision parallax collecting satellite)
has been used to measure stars 500 pc away. Distances to more than 20,000 of the nearest
stars have been measured in this way.
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Absolute magnitudes and distance
Once we have distance and apparent brightness we can get the luminosity of a star, i.e., its
intrinsic brightness. Since apparent magnitude m is in such prevalent use, an analogous
measure was created called “absolute magnitude” that is directly related to luminosity.
The absolute magnitude is how bright a star would be if placed 10 pc away from the sun. If
we know the distance, the absolute magnitude M is related to the apparent magnitude m by
M = m – 5log(d/10)
(5)
where d is in parsecs. Note, if d = 10 pc, log 1 = 0, and M = m. If the star is within 100 pc or
so, we can determine d from parallax and get M directly using (5).
For example, the apparent magnitude of the full moon is -12.74.
7. Estimate the apparent magnitude of the moon at first quarter [hint: recall for each factor of
2.5 in brightness is shift of “1” in magnitude]
__________________
8. Find the distance from the moon to earth in parsecs ___________________
9. Find the absolute magnitude M of the full moon from (5) ______________
10. Calculate M using (5) for each star in table II and enter into the table.
Stars Beyond Parallax Measures
If stars are too far away to give us d from parallax, we can rearrange (5) to find d from M
d  10( m  M )/51
(6)
again, d is in parsecs. So, if we know M we can find d.
But, how do we find M independently of d ? It turns out luminosity (and thus M) can be
estimated from a star’s color and its composition. Both of these fall into the realm of
“spectroscopy” – how light intensity varies with wavelength or frequency.
In general, an object’s luminosity is due to both its surface area and its temperature.
Luminosity is proportional to the fourth power of the temperature (the Stefan-Boltzmann
law) and the temperature can be obtained from the spectrum of the star (how the intensity of
starlight varies with wavelength). In fact, in terms of a star’s surface temperature T,
luminosity L, and radius R we can write
2
4
 R  T 
L
   L
 R  T 
4
(7)
where the subscript “” refers to the sun.
This relationship is utilized in the Hertzsprung-Russell (HR) diagram. The HR diagram
locates stars and other stellar objects on a graph of luminosity (or M) vs temperature (or
spectral type).
11. Go through this tutorial on the HR diagram. At the end of the tutorial you are asked 4
questions. After you output the answers for “printing”, you can copy them below
1)
2)
3)
4)
Look at this animation of stellar evolution in an HR type diagram (where a scale of
luminosity is in units of luminosity of the sun). Answer 12-16.
12. List the 8 stages shown in the star’s lifecycle_____________________________
_____________________________________________________________________
13. What is the hottest stage? ________________________________
14. What is the most luminous stage? ________________________________
15. What is the most long-lived stage? ________________________________
16. What is the coolest stage? _____________________________
Putting stars on the HR diagram
Use this link to do the remaining questions. It is an interactive HR diagram. It enables you
to select a position (denoted by the red “x” on the right page) in the HR diagram where a star
might be. By default, the “main sequence” line is shown in red as well as the “isoradius
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lines” in green. Note you can move the position of the red “x” with the mouse, and when
you do that the luminosity and temperature on the left page changes, and vice versa.
In the “options” of the right page select the check box “luminosity classes”. You see the 4
basic colored regions: the green “main sequence” of “dwarfs” bounded by the main sequence
line, the blue “super-giants” region, the red “giants” region and the “white dwarfs”.
17. The most luminous stars are in which region? _________________________
18. The least luminous stars are in which region? __________________________
In “Plotted Stars” select “the brightest stars”. You see a bunch of stars populating the
diagram as dots. Most are dwarfs, some are blue giants, some are red giants. Select the
“nearest stars”. They are all cooler stars – our sun is about 5800 K – including mostly “red
dwarfs”.
Now, on the right page select y-axis scale for “magnitude”. You can see the HR diagram is
expressed with a scale “Absolute Magnitude” on the vertical axis. So, absolute magnitude is
just another way of talking about luminosity.
Note than when you click on the “red x” in the right hand page and move it, the position
changes, but also the temperature, luminosity and radius value on the left page, as well as the
star size compared to the sun, changes.
Now, use the “absolute magnitude” y-scale, “temperature” for the x-scale, and the selections
“luminosity classes”, “show main sequence”, “show isoradius lines” for the following
question:
19. For each star in table II, position the red “x” on the right page to match up with the
absolute magnitude and temperature you obtained in table II. Enter what region/type of star
it is (i.e., one of main (dwarf), blue giant, red giant, red dwarf, or white dwarf) and enter
into the table under “HR region”. Also, note the radius R on the left page and put that into
the table.
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