To plot the H-R diagram for stars, and use it to estimate the temperature and luminosity of a star, given its spectral class. To calculate the mass, radius, volume, density and lifetime of a star, using the appropriate equations and graphs.
Calculator and semi-logarithmic graph paper. The later is provided at the end of this exercise.
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. Life-time and chemical composition
Let us examine how each of these quantities can be deduced.
Photos pherlc Temperature
The photospheric temperature (T) is measured in terms of 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, which states that ~ = 2.898 x 10 ~ inK! Am~ where the maximum wavelength is measured in meters.
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 0, B, A, F, G, K, M, with the 0 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 | 50,000 K |
| BC | 25,000 K |
| AC | 10,000 K |
| FO | 8,000 K |
| GO | 6,000 K |
| KO | 5,000 K |
| MO | 4,000 K |
| M7 | 3,000 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. Here are the steps by which that calculation is done:
The parallax angle of the star is measured. The distance (d) is calculated.
The apparent visual magnitude (in) is measured.
The distance (d) is used to find the distance modulus, M - in, where M is the absolute visual magnitude.
The apparent visual magnitude (in) and distance modulus (M - in) are used to calculate the absolute visual magnitude (M), since M - rn = 5 — 5logd.
The luminosity (L) is calculated from the absolute visual magnitude (M), using the equation, L = 85.51 x 10.O.4M where L is measured in solar units. This means 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 150-200 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 six steps from above (in reverse order). Estimating the distance of a star in this manner is called spectroscopic parallax.
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 binaty 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 accurately estimate the mass. Today astronomers call this the Mass-Luminosity Relationship, again only valid for Main Sequence stars. A graph between the mass and luminosity is shown at the end of the lab. Thus if a star's luminosity is calculated to be 1000, from the graph it can be seen that its mass will be 7 solar masses, or 7 times the mass of the Sun.
Radius & Volume
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, 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 8000 K, its value is (8000/5800) = 1.38 times that of the Sun. The number 1.38 rather than 8000 will be used in the equation above.
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 demoted by p can be determined, since density p = mass/volume. Since the mass and volume of the star was 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 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 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 lifetime 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 Kelvin.
2. From the H-R diagram, use the spectral class to estimate the luminosity, L.
3. Use the Mass-Luminosity Relationship (graph) 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 formula R = [L]112 fT2.
5. Calculate the relative volume using the formula V = R3.
6. The relative density can be determined using p = M/R3.
7. The estimated duration of the hydrogen-burning phase T = M/L.
Each of the quantities T, L, M, R, V, p and T will be expressed in solar units, meaning in comparison to the Sun. For the Sun these quantities are:
T(sun) = 5,800 K
L(sun) = 4x10^26 Watts
M(sun) = 2x10^30 kg
R(sun) = 7x10^8 m
P(sun) = 1400 kg/m^3
V(sun) = 1.4x10^27 m^3
T(sun) = 10^10 years