Distance to the SMC (1)
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1010
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Astronomy
Date
Dec 6, 2023
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docx
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Uploaded by gabelalex8
Determining the Distance to the Small Magellanic Cloud (SMC) Using the
Period/Luminosity Relation of Cepheid Variable Stars
The Experiment:
You are tasked with determining the distance to the Small Magellanic Cloud (a dwarf galaxy
orbiting the Milky Way) by observations of 4 Cepheid variable stars found in the Small
Magellanic Cloud (SMC).
Background:
One of the most difficult to measure, yet exceedingly important property of any object in space
is its distance. While parallax can be used to determine the distance to very close objects, it fails
very quickly as we look further into the universe. Astronomers use a device called “standard
candles” to determine distances outside the range of parallax. A standard candle is an object for
which we know exactly how much light it is actually giving off – i.e. we know its luminosity
(absolute magnitude) through some means. If we know its luminosity, and can measure how
bright it is (apparent magnitude), then we can apply the distance modulus and determine its
distance. This is how we can determine the distances to other galaxies; we find one of these
special standard candles within the galaxy and once we know the distance to it, we know the
distance to the galaxy.
One of these special standard candles is what we call Cepheid variable stars. A variable star is a
star which changes its luminosity over time, usually with a regular period. Cepheids are unique
in the fact that it was discovered that the regular period of a Cepheid is
directly related
to its
luminosity.
That is, every Cepheid star which pulsates with a certain period, P will have a
corresponding luminosity, L.
This property is very useful for astronomy. We can easily measure
the period of the star’s pulsation based on its changing brightness, and from that determine the
luminosity of the star, and by association – its distance. The only missing step here is to calibrate
this system with Cepheids close enough to get a parallax distance.
The Procedure:
First, we need to determine our calibration equation. Once we have that equation, we can plug
in the period of any Cephied variable in order to determine its corresponding absolute
magnitude. In order to create our calibration equation, we first need a list of stars for which we
know their distance through parallax – thus we know their absolute magnitude. Paired with the
logarithm of these stars’ period of variation, we can determine a relationship. A table of such
stars is provided in the Excel document, “Cepheid Period Magnitude”.
Open the Excel file, highlight the
only the numbers
, and go to “Insert” to create a Scatter plot
(no lines) in Excel of
log(P)
on the x-axis and
M
on the y-axis. Then, right-click on a point and
“Add Trendline” to fit a linear trendline to your data, making sure to check the box for “Display
Equation on chart”.
Save your graph.
This trendline equation is your calibration equation and will
have the form of:
Where
a
is the slope and
b
is the intercept,
M
= y and log(
P
) = x
1. What is your equation from the graph? This should be a linear equation of the form y = mx +
b.
y=-2.9091x-1.1994
2. Rewrite the equation of the form M = a log(P) + b, substituting your values of a (m in the
above equation) and b from your graph’s equation. Be sure to include the negative signs.
M = -2.9091log(P) - 1.1994
Equipped with this calibration equation, we can now determine the absolute magnitude of any
Cephied variable star by simply measuring their variation period, taking its logarithm and
plugging the result into the equation.
Since we are trying to determine the distance to the SMC, we need some Cephied variable stars
which reside in it. Light curves for 4 such stars are provided on the last page. For each of these
stars, use a straightedge to draw lines to each axis to determine the median apparent magnitude
(m) and the period in days (P). The median apparent magnitude is the average of the maximum
and minimum apparent magnitude, and the period is the time between two successive peaks.
When you find these values, fill in the table below. Then find the log(P) by simply taking the
base-10 logarithm of your measured period.
Period, P
(days)
log(P)
(calculate)
m, median
apparent
magnitude
M (calculate
from equation
above)
Distance
(pc)
Star 1
42
1.62
13
-5.62
53024.02
Star 2
37
1.46
13.5
-5.48
62520.26
Star 3
15
1.15
15
-4.47
78226.61
Star 4
11
1.06
14.5
-4.12
52933.94
Next, use your calibration equation M = a log(P) + b with the a and b values you got from your
graph to calculate the Absolute Magnitude, M, of each star. Fill these values in the table above.
Once you have applied your calibration and determined the Absolute Magnitude (M) of your
stars, you can apply the distance modulus equation to determine the distance to your 4 stars.
Include your calculated distances for your 4 stars in your table.
Distance modulus is:
M
=
m
– 5 log(
D
) + 5
Which can be rearranged to:
(note: this is 10 raised to a power, not multiplied by)
3. Take the average of the four distances and use it as the distance to the SMC.
Distance = 61676.2075
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