astronomy assignment

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The Luminosity of the Sun – due Oct 4

if we observe on Sep 27

This activity including the homework will be worth a total of 35 points.

READ AT LEAST PAGES 1-4 OF THE HANDOUT BEFORE THE LAB SESSION!
Make sure you understand 1) the quantities that will be measured, and 2) what a

photometer

is. You will need to
use the quantities that you measure in the last section of the lab that you will do outside of class as homework

During the lab session, you will make some measurements. All of your data should be recorded in the appropriate
places on sheet given out during the lab . Afterwards, you will be asked to answer relevant questions and perform
some calculations using the data. Please place your answers in the spaces provided in the handout. You will share
data with members of your observing group but all calculations and answers to questions should be done by you
alone.

Checklist for work to be completed during the lab session: (order of the activities is not important)

1) With members of your observing group, set-up your light bulb, baffle, and paraffin block facing the Sun as
sketched in Figure 4. Page 3 gives detailed instructions.
2) With members of your observing group, make the measurements to fill in the table on the data sheet passed
out at the start of the lab.
3) Look through the telescopes that are set up just outside the lecture hall. Answer questions and make the
drawing on the data sheet.
4) Listen to a mini-lecture on the Sun in the Lecture Hall

Due date for turning in the lab assignment is one week after the lab session (October 4, if it is clear on
September 27 ).

1. Introduction

The temperature of the Earth is determined by the luminosity of the Sun. Although Earth has a hot molten core,
the Sun contributes 99.98% of the energy that heats our planet. The next largest heat source (< 0.02%) is the decay of long-lived radioactive isotopes in Earth's interior. If the Sun were to turn off suddenly, the Earth's surface temperature would drop by ~ 250oC and all the water on Earth would turn to ice. On the other hand, if the Sun were much brighter, or much closer to the Earth, solar radiation would raise the planet's surface temperature and all our rivers, lakes and oceans would vanish in a puff of steam. The total amount of energy that the Sun radiates into space per second is called the solar luminosity (Lsun), which is measured in energy per unit time, or Watts, just like a light bulb. The solar energy reaching the Earth is measured by the solar constant, FSC. The solar constant is defined as the total amount of solar energy that enters the top of the Earth's atmosphere per unit time per unit area, with an average distance of the Earth from the Sun of

“The Sun, with all the planets revolving around it, and depending on it, can still ripen a bunch of grapes as
though it had nothing else in the Universe to do.”
–Galileo Galilei

IF THE WEATHER IS BAD ON THE DAY OF THE LAB SESSION (Sep 27),
we will have a lecture instead. The lab session would then occur at a later date, to be announced.

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Figure 1: Photometer Components

1.5 X 108 km or 1 astronomical unit (1 AU). The solar constant is measured in units of energy per unit time per
unit area, or Watts per square meter (W/m2).

The solar constant is not a true constant because the luminosity of the Sun fluctuates by a tenth of a percent or so.
Scientists who study the Earth’s climate and the greenhouse effect continuously monitor the solar constant
because changes in Fsc cause changes in the temperature of the Earth’s atmosphere. Although the changes in the
solar constant appear to be very small, only by monitoring it can we learn whether changes in the Earth’s
temperature are due to the changes in amount of solar energy we receive or some other cause.

Both the solar constant and the solar luminosity are fixed by the total amount of energy produced by
thermonuclear reactions in the core of the Sun, so they can be related by a simple formula. Imagine a sphere with
a radius of Dsun = 1 AU, centered on the Sun. The solar constant is the number of Watts that pass through each
square meter of this sphere every second. All of the energy radiated by the Sun must pass through our imaginary
sphere, which has a surface area of 4S Dsun

2, so

2 (1)4
sun

SC
sun

L
F

DS

Therefore, the total amount of energy produced by the Sun each second (that is, the solar luminosity, Lsun) can be
determined by measuring the solar constant (Fsc), knowing the distance from the Earth to the Sun (Dsun).
Compare equation (1) to that near the end of the “Light” lecture.

During the lab activity you will measure the solar
constant and then calculate the solar luminosity in the
homework section. You will determine the solar
luminosity by comparing the power output of the Sun
to the known power output of a light bulb. Near the
end of this lab, with help from Einstein’s relation
between mass and energy, E = mc2, you will calculate
the rate at which nuclear reactions in the core of the
Sun consume mass. The total mass of the Sun is 2.0 X
1030 kilograms (or 2.0 X 1027 tons!), of which about
10% can sustain these reactions. You can predict the
lifetime of the Sun by calculating how long it would
take the Sun to burn all of this mass. It may come as a
surprise to you that even with such simple experiments you can learn about processes occurring in the center of
our Sun!

2. How We Use a Photometer to Measure the Sun’s
Luminosity
A photometer is a device to measure the apparent brightness of a source
of light, by detecting the amount of light falling on the device. In this
experiment, we will use a simple photometer made with two blocks of
paraffin wax and which uses your eyes as the detectors. The two blocks
are separated by a sheet of aluminum foil and held together by rubber
bands. (See Figures 1 and 2.) These photometers will be supplied to you
already assembled.

When a light source shines on one side of the photometer, the paraffin
block on that side glows because it absorbs and scatters the light shining

Figure 2: A completed

photometer

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on its surface. The closer the light is to the photometer, the brighter the wax block will appear to be. The
apparent brightness (F) of the paraffin block is determined by the inverse square law

2 (2)4
kL

F
dS

where d is the distance from the light source to the photometer, measured in meters, L is the luminosity of the
light bulb in Watts and k is a factor that account for absorption in the paraffin.

The block on the opposite side remains dark because the light is reflected by the aluminum foil and cannot pass
through. (See Figure 3.) If another light source is placed on the opposite side of the photometer, then that side will
also glow and the apparent brightness of that block can also be calculated using the inverse square law.

Now suppose that we want to measure
the luminosity of Sun and we know
how far it is from the photometer. The
side of the photometer that faces the
Sun will glow. If we place a light
bulb with a known luminosity on the
other side, then that side of the
paraffin block will also glow, but not
with the same brightness. However, if
we move the light bulb so that the
apparent brightness of both sides of
the photometer are the same, then we
can calculate the luminosity of the
Sun. (See Figure 4.) We are in effect
using the photometer to compare the
brightness of the Sun to a light bulb
with known output.

When the two sides of the photometer glow with the same brightness, the apparent brightness of the paraffin on
the light bulb side is equal to the solar constant, or F = Fsc, where F is the apparent brightness from the light bulb
and Fsc is the apparent brightness of the Sun. Substituting the inverse square law into this equation we find that

Bulb sun
2 2

sunBulb

L L
(3)

4 D4 d

SS

where LBulb and Lsun are the luminosities of the light bulb and the Sun, respectively, and dBulb and Dsun are the
distances of these two sources from the photometer. Assuming the paraffin blocks on either side of the aluminum
foil are similar, the factor, k, cancels out. We already know LBulb and Dsun, and we can measure dBulb, so this
equation can be manipulated to give the luminosity of the Sun, Lsun.

The experiment works better if the paraffin block is partially surrounded by a baffle to keep stray sunlight (mostly
reflected from the roof) from reaching it. You can use pieces of black cardboard to make a baffle, as indicated in
Figure 4.

Figure 3. The side of the paraffin facing the light glows.

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Figure 4. Comparing the brightness of the Sun to a 300-Watt light bulb.

Name:_____________________________

3. Measuring the Sun’s Luminosity in Watts

1. This is an outside activity. Each group of students will be given a 300-watt bulb, an extension cord, a
photometer, a ruler, and a large piece of black poster board.

2. Turn the light bulb on.

3. Hold the photometer
between the Sun and the
bulb with the bulb’s
filament parallel to the
face of the photometer.
Hold the other face of
the photometer towards
the Sun (see Figure 4).
One of you should hold
the cardboard baffle
around the paraffin
block. Try to point the
face of the paraffin

block directly at the Sun.

4. Move the photometer toward and away
from the Sun, stopping when the two sides
have the same apparent brightness. The
temperature difference between the Sun and
the bulb will cause a color difference on either
side of the paraffin. Just keep in mind that the
apparent brightness on both sides of the
paraffin should be similar rather than color.

5. Hold the photometer steady while a
member of your group measures the
distance in centimeters between the bulb
filament and the aluminum in the
photometer. Assume that the filament is at
the center of the bulb. Make observations as
carefully and consistently as you can.
Record each measurement on the data sheet
passed out in class (also shown at left).
Record the weather conditions.

6. As a check on your first value, repeat the above step four more times, switching responsibilities with your
group members each time. Also, try reversing the paraffin block so the sun and light bulb shine on the opposite
sides from where you started. Again, record your measurements in the table at left. If there is a major discrepancy
between any of the measurements (more than a couple of centimeters), then additional measurements are needed.
On the other hand, if the five measurements are fairly close in value, average them to yield a single measurement.

Table 1: Photometer measurements.

Bulb distance, dBulb (cm)

Measurement #1

Measurement #2

Measurement #3

Measurement #4

Measurement #5

Average Value =

Weather Conditions: (circle best choice)
A. Clear
B. Some thin clouds away from the Sun
C. Some thin clouds covering the Sun
D. Some thick clouds away from the Sun

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Name:_____________________________

4. Viewing the Sun Through the Telescopes

As you look through the various telescopes that will be setup near the Lecture Hall, sketch what you see on the
circle below representing the Sun. Indicate what color the feature has.

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Name:_____________________________

Print out this page and the following three pages. Please turn in these four pages as well as your data sheet and
drawing from the lab.

This portion of the lab is to be done on your own after class.

Answer the questions below. You can get views of the sun over the internet at the following URLs:

http://coolcosmos.ipac.caltech.edu/cosmic_classroom/multiwavelength_astronomy/multiwavelength_museum/sun.html

and some individual examples at:

http://sohowww.nascom.nasa.gov/sunspots/ for broad band visible

http://www.bbso.njit.edu/Images/daily/images/hfull2 H alpha emission line

http://umbra.nascom.nasa.gov/images/latest_eit_171.gif for extreme ultraviolet/soft X-ray

Compare the appearance of the sunspots in the pictures at these web sites.

Use the images at these sites to answer the following questions (use the links above):

Because they are cooler than the surrounding photosphere, they are dark in the _______
a.) visible (continuum, sometimes called white light) b.) H alpha emission
c.) Ultraviolet/X-ray

Because they are the sites of high energy particles, they or the surrounding regions are bright in the _________
a) visible (continuum, sometimes called white light) b.) H alpha emission
c.) ultraviolet/X-ray

Hot gas streaming out into the corona from active regions becomes clear in the _____________
a.) visible b.) H alpha emission
c.) Ultraviolet/X-ray

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Name:_____________________________
Computing the Sun’s Luminosity and Lifetime

These calculations are easiest using scientific notation. Visit http://janus.astro.umd.edu/astro/scinote/ if you need
practice. We have also created a web site to assist you with these calculations in case you are unfamiliar with
using scientific notation on your calculator. Go to http://ircamera.as.arizona.edu/Astr170B1/Web_sheet.htm to
find it. You enter values in the yellow cells and values that correspond to some of the steps below have blue boxes
around them. One quirk of this web site is that you must enter values using the format for scientific notation used
in spreadsheets like Excel: 1×1012 would be entered as 1e12, for example. Be careful that the default values that
appear are completely overwritten by your entries. You may use the option to email your web sheet to the
professors but please transfer your results to the appropriate places on these pages (and complete the steps that are
not present on the web page).

1. Solving for the Sun’s luminosity (or the solar luminosity) requires several steps, which are outlined below.
(a.) What is the average distance between the 300-watt bulb and your photometer (in cm)? How many meters is
this? This distance in meters is the value of dBulb to use in (b.). The distance to the Sun, DSun is about 1.5 X 10

8
km from the Earth. How many meters is this? This distance in meters is the value of DSun to use in (b.). Be
careful to express these distances as a number with a unit (eg., 10 cm rather than just the number 10). You will
find it convenient to use scientific notation.

(b.) Equation (3) is reproduced below and then re-arranged algebraically to isolate Lsun which is the quantity we
are measuring:

Bulb sun
2 2
sunBulb

2 2Bulb Bulb
Sun sun sun2 2

BulbBulb

L L
(3)

4 D4 d
L L

L 4 D D
d4 d

SS

x S x
S

So LSun =
�����
�����ʹ

ൈ����ʹ

Plug in your values on the right hand side of the equation and compute LSun either using the web site or your
calculator.
Write down the equation for LSun with your values entered:

Computed value of LSun from the web site or your calculator: ________________________________________

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Name:_____________________________

2. (a.) The accepted value of the Sun’s luminosity is Lsun = 3.8 X 10

26 Watts. How does your value compare to the
accepted value (express this comparison mathematically by stating “My value is XX% greater [or smaller] than
the accepted value).

(b.) Why do you think your derived values for the Sun’s luminosity is different than the average accepted values?
(Hint #1: The light bulb filament is cooler than the surface of the Sun, so it is only 1/3 as efficient in converting
watts to visible light. Hint #2: Think about the sources of error in measurement, etc,)

3. Use your derived value of the Sun’s luminosity from Question #1 to predict how long the Sun will continue to
burn. Recall that the Sun is converting 4 hydrogen atoms to one helium atom in its core, and that a small amount
of matter is converted to energy in this process. Einstein’s famous equation, E = mc2, expresses how much energy
is produced when mass is converted to energy. A Watt is the same as 1 Joule /second and a Joule is a unit of
energy. The Sun’s luminosity can be equated to the energy produced from the rate at which mass is being
converted in the core of the Sun: Fill in the blanks in the equation below using your value for Lsun from question
1 b:

ୗ୳୬ܮ ൌ ̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴
୉୬ୣ୰୥୷
ୱୣୡ୭୬ୢ���ሺݏܽ�݁݉ܽݏ�݄݁ݐ��݁ݎܽ�ݏݐ݅݊ݑ���

௞௚�௖మ
௦௘௖௢௡ௗሻ

so doing some algebra and using c = 3×108 meters/second, compute using the web site or your calculator

ௌ௨௡ܮ
ܿଶ ൌ ̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴̴

݇݃
݀݊݋ܿ݁ݏ

(ie, divide your value for the Sun’s luminosity by the speed of light squared, web sheet can help here)

which gives the rate at which mass is being converted to energy in the Sun’s core meaning that every second
some number of kilograms are turning into energy that powers the Sun.

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Name:_____________________________

When hydrogen is converted to helium, only a small fraction of the original mass of hydrogen is converted to
energy; most of the mass remains as helium. Only 0.7% of the hydrogen mass disappears and is converted to
energy. At what rate is hydrogen participating in the conversion of H to He:

Rate of hydrogen conversion to helium =
௥௔௧௘�௢௙�௠௔௦௦�௖௢௡௩௘௥௦௜௢௡�௙௥௢௠�௔௕௢௩௘

଴Ǥ଴଴଻ = ���ቆ
ై౏౫౤
ౙమ

଴Ǥ଴଴଻ቇ ൌ�

__________________ kg/second

Only the hydrogen in the core of Sun where the temperature and pressure are sufficiently high can be converted to
helium. The core contains 10% of the mass of the Sun. The mass of the Sun is 2 x1030 kgs. Compute the mass of
the Sun’s core.

Mass of the core of the Sun = _____________________ kg

If hydrogen is being converted to helium at the rate you calculated above, how long will it take to convert all the
mass in the core of the Sun to helium? (express your answer first in seconds and then convert to years;
The number of years is the number of seconds divided by 3 X 107 seconds/year).

Time to convert all the mass in the core = Mass of the core / Rate of conversion = ______________ seconds

= ______________ years