PHYSICAL PROPERTIES OF A STAR Objective

1. To plot the H-R diagram for stars and use it to estimate the temperature and luminosity of a star, given its spectral class.

2. To determine the mass, radius, and lifetime of a star, using the appropriate equations and graphs.

Equipment Scientific calculator, pencil, highlighter, and semi-logarithmic graph paper, which is

provided at the end of this document.

Introduction There are six physical quantities, which are used to define a star:

1. Photospheric Temperature

2. Luminosity

3. Mass

4. Radius and Volume

5. Average Density

6. Lifetime and chemical composition

Let us examine how each of these quantities can be deduced.

Photospheric Temperature

The photospheric temperature (T) is measured in K. This can be calculated by direct observation from Earth. The photosphere of a star emits a continuous spectrum observable from the Earth. By dispersing the spectrum and graphing its Planck curve, the maximum wavelength can be determined using Wien's Law, T = 2.898 x 106 K-nm / λmax where the maximum wavelength is measured in nanometers.

Another method used to determine the temperature of a star is by interpreting its spectral signature. Astronomers have correlated the spectral lines observed with the degree of ionization present in the star’s photosphere. Since temperature determines the degree of ionization, once the spectral class of a star is identified, it is possible to use a table like the one below to determine a star’s temperature. Remember the spectral sequence is O, B, A, F, G, K, M, with the O stars being the hottest. Each letter category is in turn divided into 10 sub-categories, ranging from zero to nine. A star with the classification B9 is therefore slightly cooler than B8, but hotter than A0.

1

Spectral Type Temperature

O5 30,000 K

B0 25,000 K

A0 10,000 K

F0 8,000 K

G0 6,000 K

K0 5,000 K

M0 4,000 K

M7 2,000 K

Luminosity

The luminosity is the energy emitted by the star's photosphere each second at all wavelengths of the electromagnetic spectrum. If the distance to the star can be determined, the luminosity can be calculated. Here are the steps by which that calculation is done:

1. The parallax angle of the star is measured.

2. The distance (d) is calculated.

3. The apparent visual magnitude (m) is measured.

4. The apparent visual magnitude (m) and distance modulus (m – M) are used to calculate the absolute visual magnitude (M), since m – M = 5 log (d) – 5.

5. The luminosity (L) is calculated from the absolute visual magnitude (M), using the equation, L = 10-(M-4.83)/2.5 where L is in solar units. The absolute visual magnitude (M) of the Sun is 4.83. This means that if the value of L works out to be 5, the star is 5 times more luminous than the Sun.

Unfortunately stars that are further than 200 pc are too far away for their parallax to be measured. The luminosity for these stars has to be estimated using other techniques.

The luminosity of a hydrogen-burning, main sequence star can be estimated using the H-R Diagram (i.e., luminosity-temperature plot) which does not require knowing the distance. As a matter of fact once the luminosity is estimated from the H-R Diagram, the distance can then be calculated using the six steps above, but in reverse order. Finding the distance of a star in this manner is called the spectroscopic parallax method.

Mass

The mass of a star is a measure of how many and what types of atoms it contains. Astronomers first measured the mass of stars in binary systems (i.e., systems that contain two stars gravitationally bound to each other). Approximately 50% of the stars are members of binary systems.

2

For nearby systems with a measured parallax and known distance, Newton's Law of Gravity and Kepler's Third Law of Planetary Motion can be used to calculate the total mass of the stars in these systems. Further observations of the two stars as they orbit about each other can be used to calculate each of the two masses.

Of course, not all stars are in binary systems, and not all binary systems have a measurable parallax. When astronomers compared the masses and luminosities of hydrogen-burning, main sequence stars, they discovered that the luminosity could be used to estimate the mass accurately. Today, astronomers call this the Mass-Luminosity Relationship, which is only valid for main sequence stars. A graph between the mass and luminosity is shown below. Thus if a star’s luminosity is calculated to be 1,000, from the graph below it can be seen that its mass will be 7 solar masses, or 7 times the mass of the Sun.

Radius & Volume

The luminosity represents the total energy output of the star per second. This is related to the star’s temperature as noted above. But it is also related to the size of the star. A larger star will naturally have a higher energy output than a smaller one at the same temperature. Since stars are assumed to be spherical, it is possible to relate the luminosity (L) and temperature (T) of a star to its radius (R), through the equation

R = [L]1/2 / T2

In the equation above, the luminosity and temperature must be expressed in solar units. This means if you determine the real temperature of the star to be 8,000 K, its value is (8,000/5,800) = 1.38 times that of the Sun. The number 1.38 rather than 8,000 will be used in the equation above.

3

The volume compared to that of the Sun, i.e., the star’s relative volume will be V = R3.

Density

Once the mass and volume of an object are known, its density denoted by D, can be determined, since density D = mass/volume. Since the mass and volume of the star is determined relative to the Sun, the use of this equation provides the relative density, i.e., the density of the star in comparison to the Sun.

Lifetime and chemical composition

The Sun is a hydrogen-burning, main sequence star. Its chemical composition is believed to be representative of the composition of other main sequence stars:

Element % of the Total # of Atoms % of the Total Mass

Hydrogen 91.2 71.0

Helium 8.7 27.1

Others 0.1 1.9

In fact, there are some stars with far less "Others" (generally referred to as the Metals) than the Sun. These stars are found to be several billion years older than the Sun. Astronomers believe that these stars formed early in the development of the Universe when there was only hydrogen and helium. As they aged and shed their atmospheres, they deposited metals back into the Universe which were at one time hydrogen and helium.

The Sun formed out of this redeposited material. This means the atoms that make up the Sun and the planets were at one time in the interior of stars that long ago shed their atmospheres. The Sun is said to belong to Population I (i.e., the stars that formed from the redeposited material). The earlier stars are said to belong to Population II. Note: You would think these two numbers are reversed; however, astronomers identified these populations before they understood what caused their differences.

How long a star will burn depends on how much mass it has to begin with. The more mass it has, the longer it can remain "alive". But how fast it burns its fuel also plays a role. If its luminosity is high, it will be using up large amounts of fuel very fast. In that case, it will not last very long, like the journey time of a "gas-guzzling" automobile. The star’s life is thus inversely related to its luminosity and directly related to its mass.

To calculate the star’s time on the main sequence, use t = M/L where M = stellar mass and L = stellar luminosity. Once again, since M and L are in solar units, the star’s life time will also be in comparison to the Sun.

Summary

For a hydrogen-burning, main sequence star, the following procedure can be used to determine its physical quantities:

1. Read the spectral classification of the star and estimate its temperature (T) in Kelvin

4

2. From your H-R diagram, use the spectral class to estimate the luminosity (L)

3. Use the Mass-Luminosity Relationship (graph on page 3 in this exercise) to estimate the mass (M). Note: Here M refers to mass; earlier, it referred to the absolute visual magnitude.

4. Calculate the relative radius using the equation R = [L]1/2 / T2

5. The estimated duration of the hydrogen-burning phase or lifetime t = M/L

Each of the quantities T, L, M, R, and t will be expressed in solar units, meaning in comparison to the Sun. In order to find actual values, you will need the Sun’s values. For the Sun these quantities are:

Tsun = 5,800 K

Lsun = 4 x 1026 Watts

Msun = 2 x 1030 kg

Rsun = 7 x 108 m

tsun = 1010 years

You should have enough information to complete the pre-laboratory exercise.

PHYSICAL PROPERTIES OF A STAR PRE-LAB ANSWER SHEETFILL IN THE BLANK FOLLOWING EACH QUESTION WITH THE LETTER CORRESPONDING TO YOUR ANSWER.

PRE-LABORATORY QUESTIONS 1. Which of the following stars will be the hottest?

a) O9 b) A5 c) A7 d) K4

ANSWER _______

2. Which of the following stars will be the coolest?

a) G2 b) G8 c) K5 d) K2

ANSWER _______3. The temperature of a G5 star will be approximately

a) 7,000 K b) 6,000 K c) 5,500 K d) 10,000 K

5

ANSWER _______

4. Spectroscopic parallax is a method to determine

a) the spectral classification of the star. b) the parallax angle for the star c) the temperature of the star. d) the distance of the star.

ANSWER _______

5. The graph between the mass and luminosity of stars shows that

a) as the mass of a star increases, its luminosity decreases. b) as the mass of a star increases, its luminosity varies c) as the mass of a star increases its luminosity increases. d) none of the above are correct as there is no correlation between mass and

luminosity.

ANSWER _______

6. A star with a large radius will have

a) high luminosity and high temperature. b) low luminosity and low temperature. c) low luminosity and high temperature. d) high luminosity and low temperature.

ANSWER _______

7. The most abundant element present in all stars is

a) oxygen. b) metals like iron. c) hydrogen. d) helium.

ANSWER _______

8. The life-time of a star depends on its

a) luminosity. b) density. c) mass. d) both a & c

ANSWER _______

a) A star has a volume of 5 solar units. This means its volume is

b) the size of Jupiter. c) one-fifth the size of the Sun.

6

d) five times bigger than the Sun e) 25 times bigger than the Sun.

ANSWER _______

9. In the H-R diagram the two quantities plotted are a) mass and luminosity. b) distance and temperature. c) volume and distance. d) luminosity and spectral class.

ANSWER _______

Laboratory Exercise A. CALCULATING THE LUMINOSITY

1. The table below provides the apparent magnitude, absolute magnitude, distance, spectral type, and luminosity data for 15 main sequence stars. The luminosity of these stars can be found by the equation L = 10-(M-4.83)/2.5

. However, the calculations have already been done for you and the luminosities listed in the table are to be used to plot your H-R diagram.

2. Review the information given in the table below:

Star NameApparent

Magnitude (m)

Absolute Magnitude

(M)Distance

(ly)Distance

(pc)Spectral

TypeLuminosity (L / Lsun)

Acrux A 1.25 -3.9 325 99.69 B1 3100

Achernar 0.46 -2.7 140 42.94 B3 1030

Regulus 1.35 -0.58 79 24.23 B7 146

Vega 0.03 0.6 25.3 7.76 A0 49.2

Sirius A -1.43 1.47 8.58 2.63 A1 22.1

Fomalhaut 1.16 1.73 25 7.67 A3 17.4

Procyon A 0.38 2.66 11.4 3.50 F5 7.38

Alpha Centauri A 0.01 4.38 4.36 1.34 G2 1.5

Sun -26.74 4.83 1.55E-05 4.75E-06 G2 1

Tau Ceti 3.49 5.68 11.88 3.64 G8 0.46

Alpha Centauri B 1.35 5.7 4.36 1.35 K0 0.45

Epsilon Eridani 3.73 6.2 10.52 3.23 K2 0.28

61 Cygni A 5.2 7.48 11.4 3.50 K5.0 0.087

61 Cygni B 6.03 8.31 11.4 3.50 K7.0 0.041

Lacaille 9352 7.35 9.6 10.74 3.58 M2 0.013 B. DRAWING THE H-R DIAGRAMS.

7

3. Print out the semi-log graph paper at the end of this exercise. The spectral classes are marked along the horizontal (x) axis and the luminosity scale along the y axis is shown. Notice that the luminosity values in the table range from 0.013 for Lacaille 9352 to 3,100 for Acrux A. To enable us to plot this large range of numbers, we allow each large square to increase by a factor of 10. Notice that the lines on the graph paper along the vertical axis are not evenly spaced. This scale is called "logarithmic". Since the numbers increase evenly along the x-axis, the graph paper is called "semi-logarithmic".

4. Plot the luminosity versus spectral class for the 15 stars given above. These are all main sequence stars. To plot the first point for Acrux, go along the x-axis (to the right) to B, and move one more square to the right to get to B1. Then move up along the vertical (y-axis) to the line for 3,000, and go up a bit more to 3,100. Plot a point there, and write Acrux next to it. Continue till you have plotted the points for all the stars above.

5. Draw a smooth curve through the middle of the points. Do NOT join all the dots, but draw a wide curved line with a highlighter representing the "average" position. This is the main sequence line. You will need to submit the completed diagram for grading.

C. USING THE H-R DIAGRAM : Calculations for Denebola have been done for you as an example. Follow the same steps to do your own calculations for Omicron-2, Asterope, and Pi-3 Orion.

6. Use all the information we have accumulated to calculate the physical properties of Denebola (beta Leo). Its spectral classification is A3.

From the H-R diagram x-axis, you can find Denebola’s temperature is 9,400 K. Temperature (T) in Kelvin = 9,400 K Temperature (T) compared to the Sun = 9,400 / 5,800 = 1.6 (solar temp)

Draw a line from A3 up to the main sequence line you plotted and read the luminosity. It is about 20 times that of the Sun. Luminosity (L) from H-R graph = 20 (solar luminosity)

From the mass-luminosity graph on page 3, a luminosity of 20 is about 2 solar masses. Mass (M) from Mass-Luminosity graph = 2 (solar mass)

Use the equation to find Radius (R) = [L]1/2 / T2 = [20]1/2 / [1.6]2 = 1.75 (solar radius)

Use the equation to find Lifetime (t) = Mass / Luminosity = 2 / 20 = 0.1 (solar lifetime) Actual Lifetime (t) in years = 0.1 (10 billion years) = 1 billion years

Summarize your findings for Denebola, such as:

Denebola is a star of spectral class A3. Its temperature is 1.6 x more than the Sun, but its luminosity is 20 times higher. Its mass is twice and its radius is 1.75 times – almost twice the size of the Sun. Its lifetime is only 1/10th the Sun’s at only 1 billion years.

8

PHYSICAL PROPERTIES OF A STAR ANSWER SHEET

7. Do the calculations for Omicron-2 in Eridanus which has a spectral classification of K1. Show your work.

Temperature (T) in Kelvin = __________K

Temperature (T) compared to the Sun = T/5800 = _________(solar temp)

Luminosity (L) from HR graph = _________(solar luminosity)

Mass (M) from Mass-luminosity graph = _______(solar mass)

Radius (R) = [L]1/2 / T2 = [______]1/2 / [______]2 (1/2 power is the same as square root)

= [______] / [______] = (solar radius)

Lifetime (t) = Mass / Luminosity = _______ / _______ = ________ (solar lifetime)

Lifetime (t) in years = t x (1010 years) = ______ x (1010 years) = years

Summarize your findings for Omicron-2.

Omicron-2 is a star of spectral class _____. Its temperature is ______ times the

Sun, and its luminosity is _______________. Its mass is ___________ and its radius is

_________ times – about _________ the size of the Sun. Its lifetime is __________ the

Sun’s at _____ billion years.

8. Do the calculations for Asterope, which is B9 and is one of the stars in Pleiades. Show your work.

Temperature (T) in Kelvin = __________K

Temperature (T) compared to the Sun = T/5800 = _________(solar temp)

Luminosity (L) from HR graph = _________(solar luminosity)

Mass (M) from Mass-luminosity graph = _______(solar mass)

Radius (R) = [L]1/2 / T2 = [______]1/2 / [______]2 (1/2 power is the same as square root)

= [______] / [______] = (solar radius)

Lifetime (t) = Mass / Luminosity = _______ / _______ = ________ (solar lifetime)

Lifetime (t) in years = t x (1010 years) = ______ x (1010 years) = years

9

Summarize your findings for Asterope.

Asterope is a star of spectral class _____. Its temperature is ______ times the

Sun, and its luminosity is ___________ times the Sun. Its mass is ___________ and its

radius is _________ times – about _________ the size of the Sun. Its lifetime is

__________ the Sun’s at _____ million years.

9. Do the calculations for Pi-3 Orion, with spectral classification F6. Show your work.

Temperature (T) in Kelvin = __________K

Temperature (T) compared to the Sun = T/5800 = _________(solar temp)

Luminosity (L) from HR graph = _________(solar luminosity)

Mass (M) from Mass-luminosity graph = _______(solar mass)

Radius (R) = [L]1/2 / T2 = [______]1/2 / [______]2 (1/2 power is the same as square root)

= [______] / [______] = (solar radius)

Lifetime (t) = Mass / Luminosity = _______ / _______ = ________ (solar lifetime)

Lifetime (t) in years = t x (1010 years) = ______ x (1010 years) = years

Summarize your findings for Pi-3 Orion.

Pi-3 Orion is a star of spectral class _____. Its temperature is ______ times the

Sun, but its luminosity is ___________ times the Sun. Its mass is ___________ and its

radius is _________ times the size of the Sun. Its lifetime is __________ the Sun’s at

_____ billion years.

10. Summarize what you have learned from this lab. Write at least 4 complete sentences.

10

This lab was developed by MKS Publishing, Inc. - Dallas, Texas

11