Calculator & Tutorial For a simpler magnitude calculator, click here.
In the nineteenth century, the British astronomer Norman Pogson (1829  1891) proposed a system in which a difference of five magnitudes would be exactly equal to 100. Also, with more modern methods of determining brightness, some stars' magnitudes had to be adjusted to fit into the new system. So, in astronomy, brightness is expressed in magnitudes with each magnitude change equal to the 5th root of 100 or 2.5119 or a more exact figure of:
Apparent (or visual) magnitude is based upon an object's brightness as it appears from Earth.
Astronomers have chosen the distance of ten parsecs (32.6156 Light Years) as the arbitrary point at which all stars would be compared.
As is the case with absolute magnitude, luminosity is also based strictly on the object's actual (or intrinsic) brightness and the object's distance has nothing to do with it. This is similar to the way in which a light bulb's brightness is rated. A 100 watt bulb looks very bright when it is just a few feet away but if your neighbor down the street is lighting his porch with a 100 watt bulb, it looks a heck of a lot dimmer doesn't it? Nonetheless, they are both 100 watt bulbs and have equal brightness or luminosity.
Now that we have the necessary definitions, we can derive some magnitude formulas. Yes, we could easily look up these formulas, but it might be helpful to see how they are generated. (If not, you can always use the calculator.) From the definitions we learned that raising the 5th root of 100 to the power of the magnitude is the way in which brightness can be compared. For example, a second magnitude star is 15.85 times brighter than a fifth magnitude star. (2.5119³ = 15.85) We know a star's brightness as observed from ten parsecs is defined as its absolute magnitude. From the inverse square law, we know that brightness decreases by the square of the distance. Knowing these relationships, we can set up a formula:
Multiplying both sides by 10 parsecs squared we have:
taking the log_{10} of both sides:
The log_{10} (2.511886...) = .4 exactly: Multiplying both sides by 2.5 yields:
That's a much easier formula to manipulate isn't it?
5
Raising both sides of the equation to the power of ten, the left side of the equation is now 10^{log10 (distance)} which simply becomes distance. We will keep the right side equation raised to the power of ten and the equation is now:
Remember that these problems are numbered to correspond with the eight calculator buttons. 1) The star Sirius has an apparent magnitude of 1.46 and the star Regulus has an apparent magnitude of 1.36 How much of a brightness difference is this? To calculate the brightness difference we'll need to use this formula: Designating the 5th root of 100 as 'n' and raising it to the power of the magnitude difference we calculate: Rounding down to 3 significant figures, this means that Sirius is 13.4 times brighter than Regulus. 1b) If the variable star Mira has been observed from a maximum brightness of magnitude 2.0 down to an extremely faint magnitude 10.1 how much of a brightness change is that?
Rounding that to 2 significant figures makes this a brightness change of 1,700.
2) A brightness change of 30 equals how much of a change in magnitude ?
Taking logs of both sides we get:
Solving for magnitude:
Returning to our problem, we must calculate 30 brightness units in terms of magnitude.
magnitude = 1.477121255 ÷ .4 Absolute Magnitude and Luminosity are directly related but we have to see precisely how. We will have to change the brightness formula from example 1. We can use 'Luminosity' instead of the word 'Brightness' and we'll look up the absolute magnitude of the Sun which is 4.83. A star of absolute magnitude 3.83 would be about 2.5119 times (one magnitude) brighter than the Sun and a star of absolute magnitude 2.83 would be 6.31 times (two magnitudes or n²) brighter and so on. As can be seen, to calculate the luminosity, we raise the 5th root of 100 to the power of the magnitude difference and the formula is:
4) The star Altair has a luminosity ('L') of 11. What is its absolute magnitude?
Absolute Magnitude = 4.83 2.5*log_{10} (11) = 4.83 2.5*(1.0414) = 4.83 2.603 = 2.23
5) Measured from the Earth, the Sun has an apparent magnitude of 26.74 and is 4.848 x 10^{6} parsecs distant. What would be its magnitude if it were 10 parsecs away?
That number represents how much dimmer the Sun will be at 10 parsecs.
Wow, that's a lot of work. How about using the formula we generated? M = 5 +m 5 • log_{10}(distance) M = 5 26.74 5 • log_{10} (4.848 x 10^{6}) M = 21.74 (5 • 5.3144) M= 21.74 +26.572 Sun's Absolute Magnitude = 4.83
Luminosity = n^{(4.83 1.45)} = n^{(3.38)} = 22.5
To solve for apparent magnitude, we could use the inverse square law but as we saw in example 4, it's a lot easier to use the formula we generated: 5 + (m M) = 5 • log_{10}(distance) m = M + 5 • log_{10}(distance)  5 m = 1.45 + 5 • log_{10}(2.64 parsecs)  5 m = 3.55 + 5 • .422 Apparent Magnitude = 1.44
7) The star Capella has an absolute magnitude ('M') of .48 and an apparent magnitude of .08. What is its luminosity and distance?
Luminosity = n^{(4.83 M)} = n^{5.31}= 133.05 = 133
8) The star Wolf 359 has a luminosity of .00002 and is 7.78 light years (2.39 parsecs) distant. What is its apparent and absolute magnitude? M = 4.83  2.5 • log_{10}(luminosity) M = 4.83  2.5 • log_{10}(.00002) M = 4.83  2.5 • log_{10}(.00002) M = 4.83  2.5 • (4.70) M = 4.83  (11.75) Absolute Magnitude = 4.83 + (11.75) = 16.58
and for calculating the apparent magnitude: m = M + 5 • log_{10}(distance in parsecs) 5 m = M + 5 • log_{10}(2.39) 5 m = 16.58 + (5 • .3784) 5 m = 16.58 + (1.89) 5 Apparent Magnitude = 13.47 9) You are travelling in outer space and you measure the Sun's magnitude to be 2.75. How far away from the Sun are you? Solving the Sun's absolute magnitude and luminosity is easy because they are always 4.83 and 1 respectively.
As for solving for distance, we'll use: distance = 10^{((2.75 4.83 +5) ÷ 5)} distance = 10^{((2.92) ÷ 5)} distance = 10^{(.58)} distance = 3.84 parsecs = 12.5 light years
