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Stellarium Assignment 5
Assignment 5 – Star Magnitudes & Distances
Millions of stars are scattered across the sky.
Astronomers want to study these stars as carefully as
possible.
This means measuring everything we can possibly measure about them.
Unfortunately, it's not
easy to measure anything that's a septillion miles away!
As we've discussed in class, astronomers really
only measure three properties of stars:
1)
Position
2)
Brightness
3)
Color
They then use these measurements to deduce many other properties of the stars.
We've already had
practice using two methods to measure a star's position – the Altitude/Azimuth method and the Right
Ascension/Declination method.
Now it's time to look at the second stellar property:
brightness
.
Everyone agrees that stars come in different brightnesses – some are bright and some are dim!
As
discussed in class, the
brightness
of a star as it appears in the sky is given by a number called its
Magnitude
, where smaller
numbers mean brighter
stars.
Magnitudes can even be negative for very bright
stars!
Let’s explore this idea of Magnitudes.
Start
Stellarium
.
Turn off the
Atmosphere
using the Atmosphere icon at the bottom of the screen.
Click on five stars randomly.
For each star, when you've clicked on it and selected it, its
Information
will
appear, as usual, in the upper left-hand corner of the screen.
The star's
magnitude
will be listed as a
number in the 1
st
of the lines of information below the star's name.
This number is the star's
Apparent
Magnitude
.
Enter the names and apparent magnitudes of your stars below.
Click on both some
bright
and some
dim
stars to get a range of magnitudes.
Star Name
Apparent Magnitude
Polaris
2.09
Sirius
-1.09
Canopus
-0.55
Rigel
0.19
Adhara
1.41
Now let's try to figure out what the brightness of a star can tell us.
Remember, there are two kinds
of magnitudes –
Apparent Magnitude
, which is how bright the star appears to our eyes here on Earth, and
Absolute Magnitude
, which is how bright the star would be if it were ~32.6 light years away
.
Why 32.6
light years?
It's just an arbitrary number, but we'll see where it came from in a few minutes.
In class we discussed three important factors that determine a star's apparent
magnitude.
What three factors determine a star's apparent magnitude (how bright the star will appear to be in
our night sky)? Note - two are lumped together and called "Luminosity"?
The three factors that
determine a star’s apparent magnitude is how big it is (size), how hot it is (temperature), and how
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Stellarium Assignment 5
far away it is (distance). Absolute magnitude is the apparent brightness of a star if it were viewed
from a distance of 32 light-years.
Let's examine more closely
Luminosity
and
Distance
.
Luminosity
, as you remember from class,
is a measure of how much total light a star gives off every second.
It's basically the same thing as Absolute
Magnitude.
A star that looks dim to our eyes could be dim because it has a low luminosity, or
because it is far
away.
Which one is it?
If we could measure the star's distance, then we could answer that question.
PART A
Our first task is to measure a star's distance.
The most accurate way to do this is to use
parallax
.
As we discussed in class, parallax is the word we use to describe how something appears to shift its
position relative to the background when we look at it from two different angles.
In the case of stars, those
two angles are provided by looking at the star on two nights that are
six months apart
– that way we are
seeing the star from two points on opposite sides of the Earth's orbit around the Sun.
This picture helps
explain it:
Unfortunately, the stars are all so far away that the amount they shift in the sky over the course of a
year is very
small – we're talking about
arc seconds
of angle
(remember, one arc second is
1/3,600
of a
single degree)!
Astronomers have defined a distance, called a
Parsec
, which is the distance a star has to
be from Earth to shift its position by
one arc second
in the sky over the course of
six months
.
A
parsec
turns out to be
3.26 light years
.
To measure a star's distance, then, just observe that star for six months and
measure the angle it moves across the sky during that time.
This is its parallax (in arc-seconds).
Once you
know its parallax you can use the following formula to calculate its distance:
Equation 1
Where
p
is the star's parallax, in arc-seconds, and
D
is the star's distance (in parsecs
).
What the
equation shows us is that the more a star shifts its position (in other words, the greater its parallax is), the
closer it is!
Let's calculate the distances to some stars.
In
Stellarium
, search for the stars in
Table 1
below
using the
Search Window
at the top pf the screen, and when you've selected them, look in their
Information
in the upper left-hand corner of the screen.
Their
distance
is listed on the line below the
apparent magnitude (in light years)
.
For each star record its
apparent magnitude
in the second column,
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Stellarium Assignment 5
and its
distance
in light years in the fourth column. Then record its distance in parsecs. Remember, one
parsec = 3.26 light years (ly), so one light year is equivalent to approximately 0.31 parsecs. Then calculate
the parallax p in arc seconds from equation 1, making use of the star’s distance D in parsecs (to get the
parallax, p, divide 1 by D, i.e. p = 1/D). The first one, Sirius, is done for you.
Table 1
Star name
Apparent
magnitude, m
Parallax, p
(arc seconds)
Distance, in light years
(from
Stellarium
)
Distance, D, in parsecs
(using 1 light year = 0.31 parsecs)
Sirius
-1.45
0.37921
8.60
2.64
Aldebaran
1.00
0.04894
66.64
20.43
Betelgeuse
0.50
0.00655
497.95
152.67
Proxima
11.13
0.76925
4.24
1.30
Spica
0.89
0.01306
249.74
76.57
Barnard's Star
9.51
0.54725
5.96
1.83
Deneb
1.29
0.00231
1411.95
432.90
Vega
0.09
0.13026
25.04
7.68
Arcturus
0.11
0.08882
36.72
11.26
Altair
0.82
0.19496
16.73
5.13
Parallax is a powerful and relatively simple way to measure the distance to stars.
Unfortunately,
most stars are too far away for us to to measure their parallax – they just move too little in the sky to be
observed.
For these stars we will need to find another way to measure their distance.
PART B
Now let's think about the difference between
Absolute
and
Apparent
magnitude.
Apparent
magnitude measures
how bright the star appears to our eyes here on Earth
.
But some stars appear bright
simply because they're nearby.
If the stars in
Table 1
were all put at the same distance away
, say
10
parsecs
(
32.6 light years
), we would then be able to tell which star was intrinsically or naturally brighter.
No star would have an unfair advantage in being bright by virtue of the fact that it was close by!
Put a
star 10 parsecs away and then
measure how bright it appears to be, and you have that star's
Absolute
magnitude.
If the star is farther than 10 parsecs away
, then which will be a larger number - its apparent or
absolute magnitude?
REMEMBER, A LARGE MAGNITUDE MEANS DIM, AND A
SMALL
MAGNITUDE MEANS BRIGHT!
It’s absolute magnitude, because the distance will be greater
than 10 pc
If the star is closer than 10 parsecs away, which will be a larger number, its apparent or absolute
magnitude?
Apparent magnitude
Look at the distances and apparent
magnitudes of the stars in
Table 1
.
Which of these stars do
you think would have the largest
absolute
magnitude?
In other words, which star would be
dimmest if it were 10 parsecs away?
Why?
Proxima because it is the dimmest of all the stars
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Stellarium Assignment 5
Check your answer.
Look up the Absolute Magnitudes of the stars from Table 1 in Google.
Look at the stars in
Table 1
.
Which of these stars do you think would have the smallest
absolute
magnitude?
In other words, which star would be brightest if it were 10 parsecs away?
Why?
Aldebaran because it’s absolute magnitude is -0.63
Check your answer.
Look up the Absolute Magnitudes of the stars from Table 1 in Google.
Which
star has the smallest Absolute Magnitude?
Deneb
What is its Absolute Magnitude?
-7.1
So there's a connection between a star's apparent magnitude
, its absolute magnitude
, and its
distance
.
If you know any two of these three quantities, you can figure out the third!
There's a neat
mathematical way to connect
these three quantities.
Let's say
M
is a star's absolute
magnitude,
m
is the
star's apparent
magnitude, and
D
is its distance, in parsecs
.
Then the following is true:
Equation 2
If you've never seen the math term
log
before, the
log
or
logarithm
of a number is the power that you have
to raise
10
to to get that number.
In other words,
log 10 = 1
, because 10 to the first power = 10. l
og 100 =
2
, because 10
2
= 100,
log 1000 = 3
, etc.
Most scientific calculators have a
log
button.
If you don't have
one, you can use Google.com.
Just type “log” and then the number into Google.com and you’ll get the
correct value.
For each of the stars in
Table 1
, use
Equation 2
to calculate the star's
Absolute Magnitude, M
.
Enter the
results in
Table 2
below.
Remember, you have each star's
Distance,
D, and
Apparent magnitude
, m, in
Table 1
!!
Table 2
Star name
Absolute Magnitude
Star name
Absolute Magnitude
Sirius
1.4
Barnard's Star
13.22
Aldebaran
-0.7
Deneb
-7.1
Betelgeuse
-5.6
Vega
0.5
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Stellarium Assignment 5
Proxima
15.49
Arcturus
-0.3
Spica
-3.3
Altair
2.2
For one of these entries, show your work in calculating the Absolute Magnitude below:
-1.44 – 5 x log(2.6371) + 5 = -1.44 – (5 x 0.421127) + 5 = 1.45
Remember, you can check your results in Table 2 by looking up the information on Google
Which of these stars has the greatest Absolute magnitude (i.e., was the dimmest)?
Proxima
Which of these stars has the least Absolute magnitude (i.e., was the brightest)?
Deneb
There's a simple (but sometimes confusing!) connection between how bright a star really is
, how
bright it appears to be
, and how far away
it is.
Understanding that connection allows us to figure out these
three important properties of stars, furthering our understanding of the universe.
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