05Lab_Magnitudes-and-Distances
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Montgomery College *
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101
Subject
Astronomy
Date
Dec 6, 2023
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M
ONTGOMERY
C
OLLEGE
– R
OCKVILLE
A
STRONOMY
101 ASTR101
Laboratory 5 - Magnitudes &
Distances
1
Name:
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! Astronomers use measurements of
position, brightness,
and
color
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 a star’s
brightness
.
Everyone agrees that stars come in different brightnesses – some are bright and
some are dim. The
brightness
of a star as it appears in the sky is given by a
number called its
apparent 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
and
Fog
. Click on four 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 in the information below the star's name. This is the star's
apparent
magnitude
. Enter the names and apparent magnitudes of your stars below. Click
on both bright and dim stars to get a range of magnitudes, and then fill out the
table below.
Star Name
Apparent Magnitude
Sco 20
1.60
Antares
1.05
Cen 6
2.05
Sgr 3
1.75
1 Last edit Spring 2022.
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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 10 parsecs (32.6 light years) away.
Let's examine more closely
luminosity
and
distance
.
Luminosity
is a measure of
how much total light a star gives off every second. 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
. 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. The figure below 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
arcseconds of angle
(remember, one arcsecond 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 arcsecond
in the sky over the course of
six
months
.
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A
parsec
turns out to be
3.26 light years
. To measure a star's distance, then, just
observe that star for a year, and measure its parallax (in arc-seconds) during that
time, and then you can use the following formula to calculate its distance:
d
=
1
p
(
Eq.
1
)
Where
d
is the star's distance (in parsecs), and
p
is the star's parallax, in
arcseconds. What the equation shows us is that the more a star appears to shift 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
on the next page using the
Search Window
(or by simply pressing
F3
),
and when you've selected them, look in the information
given in the upper left-hand
corner of the screen. The star’s
parallax
is listed near the bottom, and the
distance
(in light-years)
is listed above that
.
For each star record its
apparent magnitude
in the column labeled “Apparent
Magnitude”, its
parallax
in the column labeled “Parallax (Arcseconds)”, and its
distance
in the column labeled “Distance from Stellarium (light-years)”. You will
fill in the first three columns of the table.
Please read the following carefully:
Stellarium
will give you the parallax in units of milliarcseconds (mas).
YOU
MUST DIVIDE THIS NUMBER
by 1000 to get it in arcseconds.
Stellarium
will give you an error bar for the parallax and the distance. You do
not need to keep the number after the “±”.
For example, for Sirius,
Stellarium
provides a parallax of 379.210±1.580 mas.
You should enter 0.37921 into the table. (The row for Sirius has been
completed for you as an example.)
Table 1
Star Name
Apparent
Magnitude
Parallax
(arcsecon
ds)
Distance
from
Stellarium
(light
years)
Distance
(parsecs)
Distance
(light-
years)
Sirius
-1.45
0.37921
8.60
2.64
8.60
Aldebaran
.85
.04894
66.64
20.43
66.60
Betelgeuse
.45
.00655
497.95
152.67
497.70
Proxima
11.00
.7685
4.24
1.30
4.24
Spica
.95
.01306
249.74
76.57
249.61
3
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ASTR101
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ABORATORY
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Barnard’s
Star
9.50
.54745
5.96
1.83
5.96
Deneb
1.25
.00231
1411.93
432.90
1411.25
Vega
.00
.13023
25.04
7.68
25.04
Arcturus
.15
.08885
36.71
11.25
36.67
Altair
0.75
.19495
16.73
5.13
16.72
Now calculate the distances to each of these stars in
parsecs
, using each star's
parallax angle and Eqn. (1)
d = 1 / p
.
Enter the results in column titled “Distance
(Parsecs)” of
Table 1
.
Finally, let’s check how well the calculated distances match with the distances from
Stellarium
.
Convert each distance from parsecs to light years, using the fact that one parsec
= 3.26 light-years. Multiply each distance in the “Distance (Parsecs)” column by
3.26
to get the distance in light-years, and then enter the result in the last
column of
Table 1
.
Do your calculated distances for the
stars (Column 6) agree with what
Stellarium
lists for each star's distance
(Column 4)?
Besides just a couple small difference
in decimal places they match up very
well
Parallax is a powerful and relatively simple way to measure the distance to stars.
Unfortunately, most stars are too far away for us 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.
Stellarium
lists the star’s
absolute magnitude
below the apparent magnitude.
The apparent magnitude tells us 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 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
.
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If the star is farther than 10 parsecs away, which will be a larger number?
Remember, the larger the number the fainter the magnitude. (check one)
-
apparent
magnitude
absolute
magnitude
If the star is closer than 10 parsecs away, which will be a larger number?
Remember, the larger the number the fainter the magnitude.
(check one)
apparent
magnitude
-
absolute
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 would have the largest magnitude. If the star were to be 10 parsecs
away magnitude considering it is the dimmest because of how far it is compared
to the other stars.
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?
The Sirius star would have the smallest magnitude since it is in the negatives it
makes it the closest. This would mean that it would also be the brightest.
There is 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:
m
−
M
=
5 log
(
d
)
−
5
(
Eq.
2
)
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If you've never seen the math term
log
or logarithm
of a number is the power that
you have to raise
10
to get that number. In other words,
log (10) = 1
, l
og (100) =
2
,
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.
With a little algebra, we can rewrite the above equation as
d
=
10
m
−
M
+
5
5
(
Eq.
3
)
Note that this is 10 RAISED TO THE POWER OF
m
−
M
+
5
5
, NOT 10 MULTIPLIED BY
m
−
M
+
5
5
!
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 (in parsecs)
and
Apparent magnitude
in
Table 1
. (Sirius has
been completed for you.)
For Sirius
:
m
−
M
=
5 log
(
d
)
−
5
M
=
m
−
5 log
(
d
)
+
5
=−
1.45
−
5 log
(
2.64
)
+
5
¿
−
1.45
−
(
5
∙
0.421
)
+
5
=
1.442
Table 2
Star Name
Absolute Magnitude (M)
Sirius
1.442
Aldebaran
-.07
Betelgeuse
-5.47
Proxima
15.43
Spica
-3.47
Barnard’s Star
13.19
Deneb
-6.93
6
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ASTR101
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Vega
0.57
Arcturus
-.11
Altair
2.2
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
Were the predictions you made (on
previous page) for the dimmest star
and the brightest star correct?
The predication was correct about the
dimmest star but not so much the
brighter star
Suppose I found a star whose absolute magnitude was -4, and its apparent
magnitude was 6. How far away is that star? (Use
Equation 3
)
For full credit,
use proper units
.
The star would be 100 Parsecs away.
Suppose I found a star whose absolute magnitude was 4.75, and its apparent
magnitude was 3.50. How far away is that star? (Use
Equation 3
)
For full
credit, use proper units
.
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This star would be 0 Parsecs away.
In your own words, describe the difference between apparent magnitude and
absolute magnitude. Explain the relationship between apparent magnitude,
absolute magnitude and distance.
The difference between apparent magnitude and absolute magnitude is that
one, apparent magnitude is the appeared brightness of the star from Earth while
absolute magnitude is used to measure how bright a star would be seen from a
standard distance which is about 32.6 light years away or 10 parsecs.
Apparent magnitude, absolute magnitude and distance all come together to
create the formula where apparent magnitude and absolute magnitude create a
relationship with the distance formula to find the distance of an object.
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