Apparent magnitude system:

 

Apparent magnitude is a measurement of flux or brightness.  Pogson is credited with modernizing the system of Hipparchus. It is defined so that a difference in magnitude of 5 corresponds to a ratio of 100 in flux.  Flux is the amount of light energy received per unit of time per unit of area.  Thus, the ratio of fluxes is related to the magnitude by:

 

f1/ f2 = 2.512 (m2- m1)

 

Apparent visual magnitude (m) is adjusted to the flux response of the human eye, and apparent photographic magnitude (mpg) is adjusted to the response of photographic film.  It is usually assumed that one is speaking of apparent visual magnitude unless otherwise stated.  A flux standard is needed for this system.  Originally Polaris was defined to have m = 2.0, but later Vega was defined as m = 0.0.  With the introduction of the more accurate color magnitude system (see below), the system has been adjusted so that m = 0.035 for Vega.  One can invert the formula above so that:

 

m2 – m1 = 2.50 log (f1 / f2)

 

The human eye, photographic film and digital cameras all have different responses at different wavelengths.  A more accurate magnitude system is needed to measure the light received at different wavelengths.  Standard broadband filters allow certain ranges of wavelengths.  Some are:

 

Filter                central wavelength                    color

U                     350 nm                         ultraviolet-violet           

B                      435 nm                         indigo-blue

V                     555 nm                         green-yellow

R                      680 nm                         orange-red

I                       825 nm                         red-infrared

 

Color magnitudes are denoted either with subscripts or the band letter:

 

mU = U

mB = B

mV = V

mR = R

mI  = I

 

This system will be used in this class.  When more than one color magnitude is measured, it is possible to extrapolate the continuous spectrum of an object.  If the object is a thermal radiator, then its temperature can be determined by two or more color magnitudes.  A color is the difference between two color magnitudes (shorter band – longer band).   Colors are related to the ratio of fluxes at different wavelengths (i.e., the slope of the objects spectrum).

 

Example colors are:

 

B – V = mB - mV   µ log (fV / fB)

 

V – R = mV - mR   µ log (fR / fV)

 

Thus, lower values for each color result from more flux at longer wavelength.  This implies that cooler objects have a larger value for color. 

 

It has been observed that typically that V color magnitude is very close to the apparent visual magnitude.  So, m @ mV is a common approximation that is sometimes not stated.  IF WE WANT ACCURACY BETTER THAN 5%, WE DO NOT MAKE THIS APPROXIMATION.

 

The current magnitude standard in the visible wavelengths is for the five color magnitudes mentioned above.  Vega (spectral type A0) has all five color magnitudes defined as zero.

 

U = B = V = R = I = 0 for Vega

 

By virtue of the definition, we find for the Sun:   mV¤ = - 26.77

Other interesting color magnitudes:

 

Alnitak (blue-white)                  B = 1.566        V = 1.697        B – V = -0.131

Betelgeuse (Red)                      B = 2.849        V = 0.769        B – V = 2.080

 

Another magnitude system is bolometric magnitude (mbol).  This measures flux at all wavelengths.  One needs two or more color magnitudes to compute a bolometric magnitude.  If the object is assumed to be a thermal radiator, then bolometric corrections can be made on the basis of a color from the difference of only two color magnitudes.  The bolometric corrections are defined as:

 

B.C.(V) = mbol – mV

 

mbol = mV – B.C.(V)

 

For reference:              mbol¤ = - 26.85           

 

 

 

 

 

 

 

 

 

Absolute Magnitude System:

 

The amount of flux we receive depends on the Luminosity (rate of light energy produced) and the distance to an object.  Stated mathematically:

 

f = L / (4pd2)

 

where f is flux, L is luminosity and d is distance.

 

If flux is related to apparent magnitude (m), then we want another magnitude sytem that relates to luminosity.  This system is the Absolute Magnitude system (M).  The absolute magnitude is equal to the apparent magnitude of an object with the same luminosity that is 10 parsecs away.  In the equations below, we assume that d is measured in parsecs (3.08 * 1016 m).  It can be shown that:

 

m – M = 5 log(d) – 5

 

d = 10 * 1.585(m-M)

 

It is important to note that an absolute magnitude system can be constructed for every apparent magnitude system (e.g., visual, color, bolometric).  The key ingredient is knowledge of the distance to an object.  If one has knowledge of luminosity (absolute magnitude) then one can determine a distance.

 

Luminosities are related by the equation:

 

L1/ L2 = 2.512 (M2- M1)

 

The luminosities (and thus absolute magnitudes) of the Sun are well studied:

 

Mbol¤ = 4.74

L¤ =  3.8 * 1026 Watts

 

We can combine this to compute luminosity in terms of the Sun’s Luminosity:

 

L  / L¤ = 10 0.4 (4.74 – Mbol)