Investigation 4-3:

Star Magnitudes


Materials


I. The Magnitude Scale

Using magnitude scale definitions on the previous page, the following  are examples of determining how many times brighter one star is than another:

A 10th magnitude object compared to a 20th magnitude object.

A 10th magnitude object is 100 times brighter than a 15th magnitude object, and a 15th magnitude object is 100 times brighter than a 20th magnitude object. So a 10th magnitude object is 100 x 100  = 10,000 times brighter than a 20th magnitude object.

A 7th magnitude star compared to a 14th magnitude star. 
A 7th magnitude object is 100 times brighter than a 12th magnitude object; a 12th magnitude object is 2.5 times brighter than a 13th magnitude object; and a 13th magnitude object is 2.5 times brighter than a 14th magnitude object. So a 7th magnitude object is 100 x 2.5 x 2.5 = 625 times brighter than a 14th magnitude object.

A 5th magnitude star compared to a 11.5 magnitude star. 
A 5th magnitude object is 100 times brighter than a 10th magnitude object; a 10th magnitude object is 2.5 times brighter than a 11th magnitude object; and a 11th magnitude object is 1.6 times brighter than a 11.5 magnitude object. So a 5th magnitude object is 100 x 2.5 x 1.6 = 400 times brighter than a 11.5 magnitude object.

A negative 5th (-5th) magnitude star compared to a 7th magnitude star is 100 x 100 x 2.5 x 2.5 = 62,500 times brighter.

Now, you try a few: How many times brighter is:

4.11. A 5th magnitude star than a 10th magnitude star?   
4.12. A 7th magnitude star than a 17th magnitude star?
4.13. A 3rd magnitude star than a 5th magnitude star?
4.14. A 3rd magnitude star than a 6.5 magnitude star?
4.15. A 12th magnitude star than a 22.5 magnitude star?
4.16. Our sun (-26 magnitude) than a 15th magnitude star?   

Ask the reverse question. Here are some examples. What is the magnitude of the star if:

It is 100 times brighter than a 15th magnitude star. A difference of five magnitudes means a difference of 100 times in brightness. Also, a lower number means a brighter star, so the star must be a magnitude 10 star.

It is 10,000 times dimmer than a 15th magnitude star.  A difference of 10 magnitudes means a difference of 10,000 times in brightness.  Also a higher number means a dimmer star so the star must be a magnitude 25 star.

It is 250 times brighter than a 14th magnitude star. A difference of 6 magnitudes: 8th magnitude.

It is 625 times brighter than a 9th magnitude star. A difference of 7 magnitudes: 2nd magnitude.

Now you try a few. What is the magnitude of a star if:

4.17. It is 100 times dimmer than a 12th magnitude star?
4.18. It is 10,000 times brighter than a 12th magnitude star?
4.19. It is 625 times brighter than a 11th magnitude star?
4.20. It is 25,000 times dimmer than a -5 magnitude star?
4.21. It is 100,000,000 times brighter than a 5th magnitude star?


II. Comparing the Magnitudes of Stars

It is a common practice in astronomy to compare the brightness of stars on the same image or on two different images. The ratio of brightness can be expressed as a difference in magnitudes.

Suppose the brightness of star1 = B1 and the brightness of star2 = B2. We could express this in magnitudes using m1 = the magnitude of star1 and m2 = the magnitude of star2.

If m1 - m2 = 1 then B2 = (2.5)1 x B1  

and if  m1 - m2 = n then  B2 = (2.5)n x B1

The following expression can be derived for the difference in magnitudes using log base 10 (which is the base commonly used by astronomers):

m1 - m2 = 2.5 log(B2/B1)     [Equation 1]

where B1 & B2 = the measured brightness values of star1 & star2

When comparing two stars on the same image, the brightness value of each star measured in SalsaJ are a measure of how many photons struck the CCD camera at the position of the star.

If m2 is know, you can rewrite the equation to solve for m1.


•    Open the image Mgclust. The brightest star on this image has magnitude, m(v) = 8.0.

•    Using the Photometry tool (), get the brightness (intensity values) of the brightest star and at least two dimmer stars.  To use the tool, select the tool and then click on the star you wish to measure.

4.22 Knowing the apparent magnitude of the brightest star, use Equation 1 (above) to calculate the apparent magnitude of each of the stars in one of your samples. A spreadsheet is one way to do these calculations.

4.23 How much brighter is the brighter dim star than the dimmest? Calculate this two ways: one based on the difference in magnitudes and one based on the ratio of Counts. These two values are probably not the same. Why not?


III. Absolute Magnitude

So far we have been dealing with apparent magnitudes, which are how bright stars appear to us on Earth. Absolute magnitude is how bright the star is intrinsically, independent of its distance away. This is related to luminosity of a star, which is the amount of energy it emits per second. 

The absolute magnitude of a star can be obtained from the apparent magnitude if the distance to the star is known. Absolute magnitude is defined to be the apparent magnitude that a star would have if it were 10 parsecs (pc) from Earth.

The apparent brightness of a star can be calculated as follows:

apparent brightness = (luminosity)/4πd2     [Equation 2]
where d = the distance to the star and 4πd2 is the surface area of the sphere over which the light is spread. 

The absolute magnitude, M, is defined as the apparent brightness of a star 10 pc away.
The apparent brightness of a star 10 pc away is:

(luminosity)/4π(10pc)2

Using the Equation 1 above:

m1 - m2 = 2.5 log(B2/B1), we get:   m - M = 2.5 log [(L/4π(10pc)2) / (L/4πd2)]

  where
  m = the apparent magnitude of the star
  M = the absolute magnitude of the star
  L  = the luminosity of the star
  d  = the distance to the star in parsecs

4.24. Use algebra and the rules for logarithms to derive the following equation, called the distance modulus, for the difference between apparent and absolute magnitude:
    m – M = 5 log (d) – 5

4.25. If a star is 2000 pc away and has an apparent magnitude of 7.0, what is its absolute magnitude?

4.26. If the star measured in Part II is 1400 pc away, what is its absolute magnitude?



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