Stellar Properties Tutorial

Measures of Distance

 Measures of Light Intensity

Flux Flux is a measure of light intensity. Flux records the light energy perunit time per unit area. Flux is measured in units of Watts per square meter. The flux of light incident on Earth coming from the Sun has a maximum value of 1360 W/m2. A common stellar flux for a bright star would be the flux for Canopus which is about 6.8 x 10-5   W/m2.

Apparent Magnitude Apparent magnitude derives from Hipparcos's system for classifying the brightnesses of stars, first magnitude being the brightest, magnitude 2 being next brightest etc. The modern system assigns the star Alpha Centauri a magnitude of 0 and for every factor of 100 less flux from a star, the apparent magnitude of the star goes up by 5. So the magnitude scale is a logarythmic scale in which brighter stars have lower magnitudes. The relationship between flux and apparent magnitude is given by a graph of Apparent Magnitude Definition

 Measures of Power Output

	Luminosity The total luminosity of the star. The Luminosity of a star is its light power output and is measured in Watts, or in solar luminosities. 1 solar luminosity ( 1 Lsun ) is the luminosity of the Sun which is ~ 3 x 1026 Watts
	Absolute Magnitude Absolute Magnitude is another way of quantifying luminosity in a logarythmic scale. The Value of absolute magnitude equals the value of apparent magnitude if the star is observed from a distance of 10 parsecs. The Sun has an absolute magnitude of 4.8. The relationship between flux and apparent magnitude is given by a graph of Absolute Magnitude Definition

 Measures of Temperature

	Temperature The effective surface temperature of the star. Temperatures are always given in the Kelvin temperature scale ( 0 K = -273 C)  This value is ~ 5800 K for the Sun.
	Lamdba max The wavelength at which the light spectrum from the star has maximum intensity. Wavelength is commonly measured in Angstroms ( 1 A =10-10 meters ) or nanometers ( 1 nm = 10-9 meters ) For the Sun lambda max is ~ 400 nm The relationship between lambda max and temperature is summarised in the graph of Wein's Law.
	Spectral Class The dark line light spectra are classified by letters according to the charcteristics of the spectral lines. It turns out that those characteristics are almost entirely explained by effective temperature, so that you can place the Spectral Classes in order of decreasing temperature O B A F G K M. Each class is subdivided into 10 subclasses by appending a number to the class name, with A0 being the hottest subclass of spectral class A and A9 being the coolest  . The temperature ranges associated with each spectral class can be summarized in a table of Spectral Classes .
	Colour Index The temperature of a star can be determined by examining the ratio of the fluxes given off at two wavelengths. A ratio of fluxes corresponds to a difference in magnitudes. Color index is defined as the difference between the blue magnitude (B) and the visual  or yellowish magnitude (V) .  The star Vega has color index 0 , it is a Class A0 star with an effective temperature of about 10,000 K. Hotter stars have negative color indexes and cooler stars have positive color indexes. The relationship between color index and temperature is summarised in a graph of colour indexes .

Radius All stars are beleived to be basically spherical in shape ( any object with a lot of gravity always is, unless it is spinning very fast .) The radius is the distance from the centre to the surface (photosphere). Stellar radius is usually expressed in solar radii Rsun = 7 x 108 meters, so a 10 solar radius star has a radius of 7 x 109 Meters.

Summary of Properties

We have listed ten properties of interest, but only five are independent : D(distance), m(Apparent Magnitude), M(Absolute Magnitude), T(Temperature), are R(Radius). Each of the other five are related to one of these five as follows.

Additional Relationships

Stephan's Law

The effective temperature of the surface of the star determines how brightly each piece of the surface area is shining ( if you were standing on it ). This means that you only need to know a star's Surface temperature and surface area to figure out its total power output or luminosity. The star's surface area is determined by its radius, and it's luminosity can be represented by its absolute magnitude Stephan's Law is represented quantitatively in the Graph of Stephan's Law.

 

The Inverse Square Law for Light

Since the intensity of light decreases as you move away from the source in a well known way ( inverse square ), to find the distance from Earth to a star you only need to compare the light intensity at Earth to what the intensity would be if we were located 10 parsecs from the star. We accomplish the comparison by forming the difference of apparent and absolute magnitudes (m-M) . If the difference is greater than zero then the apparent magnitude is greater than the absolute magnitude, so the flux is less than it would be if it was 10 parsecs away, so it must be more than 10 pc away. The inverse square Law is represented quantitatively in the Graph of Distance - Magnitude relation.

Putting it all Together

We started out with ten measureable quantities. Our five conversion relations mean that any of those ten can be directly converted to one of the five independant quantities D m M T R. Now we have discovered two relationship among those five, ( so they are not really independent, after all ). Stepan's law relates T M and R, while the inverse square law relates D M and m. These two relationships cut down the number of truly independant quantities from 5 to 3 . The upshot of this is that given the values of any 3 quantities, we can deduce the values of the other seven ( one deduction for each of our five conversions and two other relationships.)

Example #1

For instance, if we know that a star has absolute magnitude -1, apparent magnitude +1 and a radius of one solar radius ( M=-1, m=+1, R= 1R_sun,) we can start off filling in the other seven quantities 

1. Convert Absolute Magnitude into Luminosity using Graph of Absolute Magnitude Definition( M = -1 converts to about L = 200 L_sun )
2. Convert Apparent Magnitude into Flux using Graph of Apparent Magnitude Definition ( m = +1 converts to about f = 1.5 x 10^-5 W/m² )
3. From the difference m-M = (+1) - (-1) = 2, we can get the distance to the star using the Graph of Distance - Magnitude relation. ( m - M = 2 corresponds to about D = 30 Parsecs )
4. Use M = -1 and R= 1R_sun as two of the three quantities in Graph of Stephan's Law ( follow the dotted line representing R=1R_sun until you hit the vertical line representing M = -1, you then read off the temperature as being between 20,000 and 30,000 Kelvin ( a good deal closer to 20,000, so call it T = 22,000 K .)
5. Convert Temperature to lambda max using Graph of Wein's Law ( T= 22,000 K corresponds to about half way between 1,000 and 1,500 angstroms so Lambda max = 1,250 Angstroms .)
6. Convert Temperature to Spectral Class using  Table of Spectral Classes  ( 22,000 K is at the high end of the range for B type stars so we'll call it Spectral Class B2 .)
7. Convert Temperature to Colour index using  Graph of colour indexes ( 22,000 K looks to be about where the graph cuts the horizontal line representing colour index -0.25 .)