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 calculate the mass, radius, and lifetime of a star, using the appropriate equations and graphs.

Equipment

Calculator and semi-logarithmic graph paper (supplied at the end of this lab).

Introduction

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

1. Temperature

2. Luminosity

3. Mass

4. Radius

5. Lifetime and chemical composition

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

Temperature

The temperature (T) of the photosphere is measured in degrees 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 by using Wien’s

Law, T Wien = 2,900,000 nm K/  max where  max is the maximum wavelength 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 seen 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.

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 and over all wavelengths of the electromagnetic spectrum. If the distance to the star is known, the luminosity can be calculated either by using the equations for apparent brightness or absolute magnitude. The former is

Apparent Brightness = Luminosity/4 π r2 where r = distance

To use absolute magnitude the steps are listed below:

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 = 85.51 x 10-0.4M where L is measured in solar units. 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 estimated using the five steps above, but in reverse order. Estimating 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.

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 on the next page. Thus if a star’s luminosity is calculated to be 1,000 from the graph, it can be seen that its mass will be 7 solar masses, or 7 times the mass of the Sun.

Radius

The luminosity represented the total energy output of the star per second. This is related to the star’s temperature as we 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,

L = 4 π σ T4 R2 where π and σ are constants

However, since it is more meaningful to compare stellar data to that of our Sun, we can simplify the above equation as L = T4R2 where L, T and R are all expressed in solar units and the constants 4πσ have been removed as they would be the same for the Sun and another star. Rearranging this equation gives R 2 = L / T4 or

R = √ [L] / T2

It is important to remember that 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.

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 will depend 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 will also play a role. If its luminosity is high, it will be using up large amounts of its 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 t 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 degrees Kelvin .

2. From the H-R diagram, use the spectral class to estimate the luminosity (L). To do this you must draw a line vertically upwards from the x-axis until it meets the H-R graph and then draw a line horizontally until it meets the y- axis.

3. Use the Mass-Luminosity Relationship graph on page 4 to estimate the mass (M). Follow the same steps noted in 2 above, namely draw a horizontal line from the luminosity axis until it meets the plotted line and then draw a vertical line until it meets the mass axis.

4. Calculate the relative radius using the formula R = √ [L] /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. For the Sun these quantities are:

Tsun = 5,800 K L sun = 4 x 1026 Watts M sun = 2 x 1030 kg R sun = 7 x 108 m t sun = 1010 years

Laboratory Exercise

PRE-LABORATORY QUESTIONS

1. Which of the following stars will be the hottest? (a) O9 (b) A5 (c) A7 (d) K4

2. Which of the following stars will be the coolest? (a) G2 (b) G8 (c) K5 (d) K2

3. The temperature of a G5 star will be approximately (a) 7000 K (b) 6000 K (c) 5500 K (d) 10,000 K

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.

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.

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.

7. The most abundant element present in all stars is (a) oxygen. (b) metals like iron. (c) hydrogen. (d) helium.

8. The life-time of a star depends on its (a) luminosity. (b) density. (c) mass. (d) both a & c

9. A star has a volume of 5 solar units. This means its volume is (a) the size of Jupiter. (b) one-fifth the size of the Sun. (c) five times bigger than the Sun (d) 25 times bigger than the Sun.

10 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.

A. RECORDING THE LUMINOSITY

The table below provides the spectral class and luminosity for 11 main sequence stars. The luminosity was calculated from the apparent magnitude, absolute magnitude and distance using the equations given on page 2. To make it easier for you, the luminosity values determined by those calculations are shown in the table below.

Stellar data table:

Star name Spectral type

Luminosity

Acrux B3 7500

Achernar B5 3600

Vega A0 60

Sirius A1 26

Fomalhaut A3 18

Procyon A F5 8.6

Alpha Centauri

G2 1.6

Sun G2 1.0

Epsilon- Eridani

K2 0.37

61-Cygni A K5 0.17

Lacaille M1 0.05

B. DRAWING THE H-R DIAGRAMS.

11 a.

Print out the semi-log graph paper at the end of this exercise. The spectral classes will be marked along the horizontal (x) axis and the luminosity scale along the y axis must be chosen as shown. Notice that the luminosity values in the table range from 7500 for Acrux to about 0.05 for Lacaille. 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”. Choose the numbers along the y-axis as given on the graph paper.

11 b.

Plot the luminosity versus spectral class for the 11 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 three more squares to the right to get to B3. Then move up along the vertical (y-axis) to the line for 7000, and use your judgement to go up a little more to get to 7500. Plot a point there, and write Acrux next to it. Continue till you have plotted the points for all the stars above.

11 c.

Draw a smooth curve through the middle of the points. You do NOT have to join all the dots, but draw a wide line with a highlighter pen representing the “average” position. This is the main sequence line.

C. USING THE H-R DIAGRAM

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

IMPORTANT! The steps (i) – (vii) below show you how the stellar properties will be determined. All the calculations for Denebola are shown. Please work your way through this example. It will be worth the effort!

(i) From the H-R diagram x-axis, you can estimate that Denebola’s temperature is between 9,000 and 10,000 K. Let’s choose it to be 9,400 K. Temperature (T1 in Kelvin) = 9,400 K

(ii) Temperature as compared to the Sun = T1 / 5800 = 9,400 /5800 = 1.6

T = 1.6 (this value will be used in step vi)

(iii) Let’s check luminosity of Denebola from H-R graph. If you draw a line from A3 up to the main sequence line you plotted, and then read the corresponding luminosity, it turns out to be about 20 times that of the Sun. Therefore, L = 20

(Note this value may be different since it depends on where you drew your H-R line. Since we are doing approximate calculations, it is more important for you to understand the process than get the exact answer).

(iv) From the mass-luminosity graph on page 4, a luminosity of 20 corresponds to 2 solar masses. M = 2 (solar mass)

M = 2(2 x 1030 kg) = 4 x 1030 kg

(v) Radius = 2

T = √[20] / [1.6]2 = 1.75 (solar radius)

Radius = 1.75 ( 7 x 108 m) = 1.23 x 109m

(vi) Lifetime = t = Mass/Luminosity = 2/20 = 0.1 solar lifetime

Lifetime = 0.1 (10 billion years ) = 1 billion years.

12. Do the calculations for Omicron-2 in Eridanus which has a spectral classification of K1.

Temperature (T1 in Kelvin) =

Temperature as compared to the Sun = T1/5800 = T = —————————————————————————————————-

Luminosity from HR graph (L) = _____________________ in Solar Luminosities ***Remember to draw a vertical line up from K1 to the main sequence line and then a horizontal line to the luminosity axis. Show this on your graph.*** —————————————————————————————————– Mass from Mass-luminosity graph on page 4 (M) = _____________ in Solar Masses To get the mass of star in kilograms multiply M times 2 x 1030 kg = —————————————————————————————————–

Radius R = 2

T = ____________________ in Solar Radi

To get the radius of the star in meters multiply R times 7 x 108 m =

—————————————————————————————————– Lifetime (t) = M/L = _________________________ in Solar Lifetimes To get the lifetime of the star in years multiply the t times 10 Billion years =

________________ years

13. Do the calculations for Asterope, which is B9 and is one of the stars in Pleiades.

Temperature (T1 in Kelvin) =

Temperature as compared to the Sun = T1/5800 = T = —————————————————————————————————-

Luminosity from HR graph (L) = _____________________ in Solar Luminosities ***Remember to draw a vertical line up from B9 to the main sequence line and then a horizontal line to the luminosity axis. Show this on your graph.*** —————————————————————————————————– Mass from Mass-luminosity graph on page 4 (M) = _____________ in Solar Masses To get the mass of star in kilograms multiply M times 2 x 1030 kg = —————————————————————————————————–

Radius R = 2

T = ____________________ in Solar Radi

To get the radius of the star in meters multiply R times 7 x 108 m =

___________________ years

14. Do the calculations for Pi-3-Orion, with spectral classification F6.

Temperature (T1 in Kelvin) =

Temperature as compared to the Sun = T1/5800 = T = —————————————————————————————————-

Luminosity from HR graph (L) = _____________________ in Solar Luminosities ***Remember to draw a vertical line up from F6 to the main sequence line and then a horizontal line to the luminosity axis. Show this on your graph.*** —————————————————————————————————– Mass from Mass-luminosity graph on page 4 (M) = _____________ in Solar Masses To get the mass of star in kilograms multiply M times 2 x 1030 kg = —————————————————————————————————–

Radius R = 2

T = ____________________ in Solar Radi

To get the radius of the star in meters multiply R times 7 x 108 m =

___________________ years

15. Summarize what you have learned from this lab.

Rubric for online: Question 1- 10 each worth 0.5 points Question 11 – Graph – worth 3 points Question 12 -14 each worth 5 points Question 15 worth 2 point

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

PHYSICAL PROPERTIES OF A STAR

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