**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 = B_{1} and the
brightness of star2 = B_{2}. We could
express this in magnitudes using m_{1} =
the magnitude of star1 and m_{2} = the
magnitude of star2.

If m_{1}
- m_{2} = 1 then B_{2}
= (2.5)^{1} x B_{1}

and if m_{1}
- m_{2} = n then B_{2}
= (2.5)^{n} x B_{1}

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

**m**_{1
}**- m**_{2}**
= 2.5 log(B**_{2}**/B**_{1}**) **[Equation 1]

where B_{1}
& B_{2 }= 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 m_{2} is know, you can rewrite
the equation to solve for m_{1}.

• 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)/4πd^{2} [Equation 2]

where d = the distance to the star and 4πd

^{2} 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:

m_{1} - m_{2} = 2.5 log(B_{2}/B_{1}), we get: m
- M = 2.5 log [(L/4π(10pc)^{2}) / (L/4πd^{2})]

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?**