PHYS110-Fall2023-Lab11
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Astronomy
Date
Dec 6, 2023
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PHYS 110L Lab # 11
Distance to M100 as Determined by Cepheid Variable Stars
1
Instructions:
Please read and follow the steps described below and answer
all
questions. Feel
free to use Excel for your calculations.
Introduction:
Part #1 Using Cepheids as Distance Estimators
Measuring the distance to an astronomical object is very difficult and is one of the greatest
challenges facing astronomers. Over the years different distance estimators have been found.
One of these is a class of stars known as Cepheid variables that change in brightness.
Cepheids are rare and very luminous stars that have a very regularly varying luminosity. They
are named after the star Delta-Cephei in the constellation Cepheus, which was the first known
example of this particular type of variable star.
In 1912 the astronomer Henrietta Leavitt observed 20 Cepheid variable stars in the Small
Magellanic Cloud (SMC), a dwarf galaxy orbiting the Milky Way. The small variation in
distance to individual Cepheid variables in the SMC are negligible compared with the much
larger distance to the galaxy. Henrietta Leavitt uncovered a relation between the intrinsic
brightness (luminosity) and the pulsation period (time between peak brightness) of Cepheids and
showed that more luminous Cepheids have a longer period
(known as the Period-Luminosity relation). By observing
the period of a Cepheid, one can deduce its luminosity and
so, by observing its apparent brightness, calculate its
distance. This is done by recalling that brightness decreases
with the square of the distance (
e.g.,
if the distance to a star
were to double, it would look only ¼ as bright). In this way
Cepheids can be used as a “standard candle” to measure
distance. Cepheid stars can be distinguished from other
variable stars by their characteristic light curves (a plot of
brightness versus time; see Figure 1).
1
Modified from
The Distance to M100 as Determined by Cepheid Variable Stars
, from The ESA/ESO Astronomy
Exercise Series Exercise 2.
Learning Objectives:
In this lab assignment you will use observations of Cepheid variable stars to measure the
distance to the galaxy M100. You will then compare your result with the published value.
Figure 1:
The light curve of a
Cepheid has a shape with brightness
rising sharply, followed by a gentle
decline.
period
2
Part #2 The Spiral Galaxy M100
M100 is a magnificent spiral galaxy in the large Virgo cluster of galaxies. The Virgo cluster
contains approximately 2,500 galaxies. M100 is a rotating system of gas, dust, and stars similar
to the Milky Way, and is viewed face on.
M100 is one of the more distant galaxies where accurate
measurements of Cepheid variables have been made. This
lab is based on Hubble Space Telescope images and data
for this galaxy.
Measurements and Calculations:
The Period-Luminosity relation for Cepheid variable stars
has been revised many times since Henrietta Leavitt’s
first measurements. Today the best estimate of the relation is:
𝑀 = −2.78 log
10
(𝑃) − 1.35
where
𝑀
is the absolute magnitude of the star and
𝑃
is the period measured in days.
Light curves for a sample of 12 Cepheids in M100 that have been measured by the Hubble Space
Telescope are shown in Figures 3 and 4.
Table 1
Cepheid Star
Number
Period
(days)
Absolute Magnitude
(M)
Average Apparent
Magnitude
<m>
Distance
(Mpc)
1
2
3
4
5
6
7
8
9
10
11
12
Figure 2:
The M100 spiral galaxy
located in the Virgo cluster of
galaxies.
3
Step 1.
Estimate the period of each Cepheid in Figures 3 and 4 and record your answer in Table
1. Recall that the period is the time taken for the star to go from maximum to maximum
brightness or the time to go from faintest to faintest brightness.
Step 2.
Using the Period-Luminosity equation given above, calculate the absolute magnitude for
each of the 12 Cepheid stars and record this information in Table 1.
Our goal in this lab is to calculate the distance to M100. The distance equation relates absolute
magnitude (
M
), apparent magnitude (
m
), and distance (
d
):
𝑀 = 𝑚 − 5 log
10
(?/10 𝑝?)
.
Rearranging this equation and solving for distance, we have:
? = 10
0.2(𝑚−𝑀+5)
where the distance,
d
, is in parsecs (pc).
To calculate distance using the above equation, we need both absolute magnitude (
M
) and
apparent magnitude (
m
). Since the apparent magnitude changes with time (see Figures 3 and 4),
astronomers use the average apparent magnitude (<
m
>) in the distance equation. The average
apparent magnitude can be found by adding together the maximum and minimum apparent
magnitudes from the light curve and divide the sum by 2 to get the average value.
Step 3.
Calculate the average apparent magnitude (<m>) for each Cepheid and record your
values in Table 1.
Step 4.
Use the distance equation and calculate the distance to each Cepheid. Convert your
distances (in pc) to megaparsecs (Mpc) by dividing your answer by 1 million. Record the
distances in Mpc in Table 1.
Question 1:
Why is the distance not exactly the same for each Cepheid variable star?
Because the apparent magnitude and absolute magnitude are different for each variable star.
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4
Question 2:
Could the fact that the 12 Cepheid stars have different positions within M100,
which has a diameter of 33 kpc (33,000 pc), be the reason why the distances vary?
Yes, because of dust, gas could affect the distance between positions or technical issues.
Question 3:
What is the average value of the distance for the 12 Cepheid variable stars? Please
include the unit for your distance.
21.66391
Question 4:
In the original scientific paper using Hubble Space Telescope measurements, the
distance to M100 was calculated as 17.1 ± 1.8 Mpc (where 1.8 Mpc is the uncertainty in the
distance). How does your average distance for M100 from Question 3 compare with this
published value?
The paper has =17.1 +1.8 or 17.1-1.8 but we got = 21.66391. The difference would be 21.66391
–
17.1 =
4.563912
5
Figure 3:
Light curves for Cepheid variables in M100 that have been observed with the Hubble Space Telescope.
The absolute magnitude,
M
, is determined from the period of the Cepheids.
6
Conclusion.
Please provide feedback regarding the lab assignment. Are there things that you
liked or disliked? Thanks!!
Figure 4:
Light curves for Cepheid variables in M100 that have been observed with the Hubble Space Telescope.
The absolute magnitude,
M
, is determined from the period of the Cepheids.
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