1. Stars: Distances & Magnitudes
Chapter 3
Stars: Distances & Magnitudes
 Nick Devereux 2006
Revised 8/2012

2. Astrophysical Units & Constants
In addition to the usual list of physical constants – listed in
Appendix A (pg A-2), there is another list of astronomical
constants that we must be familiar with and these are listed in
on page A-1.
 Nick Devereux 2006

3. The Sun as the “yardstick”
Since the distances and masses are so large in astronomy,
the basic units of measurement are expressed in terms of the
Sun.
The Astronomical Unit (AU) is the distance between the Earth and
the Sun,
1 AU = x 1011 m
Which is perhaps more familiar to you as 93 million miles.
 Nick Devereux 2006

4. The parsec for larger distances, there is the parsec (pc)
1 pc = x 1016 m
 Nick Devereux 2006

5. Angular units
Astronomers can measure the angular extent on the sky for
celestial objects, even if they don’t know how far away they
are, and therefore unable to attribute a linear size.
Radians
360o = 2 radians
So, o = 2  /360 radians
or, o =  /180 radians
 Nick Devereux 2006

6. Angular Units (continued)
Arc seconds (‘’) and Arc Minutes (‘)
1‘ = 60 ‘’
1o = 60 ‘ = 3600 ‘’
From previous page…
1o =  /180 radians
So,  /180 radians = 3600 ‘’
 /(180 x 3600) radians = 1‘’
Or, ‘’= 4.8 x 10-6 radians
and ‘’ = 1 radian
 Nick Devereux 2006

7. Getting back to the pc…..
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8. The pc (radian) = arc /radius 1‘’/ 206265 ( ‘’/ radian) = 1 AU / 1pc
So, 1 pc = 1 AU x
1 pc = x 1011 x m
Or, pc = x 1016 m
 Nick Devereux 2006

9. In words, a parsec is the distance at which the separation between the Earth and the Sun could be resolved with a medium sized telescope…
 Nick Devereux 2006

10. What does resolved mean?
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11. The Resolving Power of a Telescope
Depends on both the size of the telescope mirror, D,
and the wavelength,  ,of the light under observation.
(radian) = 1.22  /D
with  and D in the same units.
For the Hubble Space Telescope D = 2.4m
and  =  = 5500 Å .
 Nick Devereux 2006

12. Question: How far away would you have to hold a dime (2cm in diameter) for it to subtend an angle of 1‘’, 0.l‘’ ?
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13. Mass Usually, the masses of stars, galaxies, clusters of galaxies
are given in terms of the mass of the sun,
1 M  = 1.99 x 1030 kg
 Nick Devereux 2006

14. Measuring Brightness Brightness is measured in a variety of ways;
eg. Magnitude, Flux, and Luminosity
 Nick Devereux 2006

15. Luminosity of the Sun The luminosity of the Sun, 1 L  = 3.9 x 1026 W
You may recall that Watts = Joules/sec, so the luminosity of the Sun is a measure of the rate of flow of energy through the surface of a star.
Concept: Think of luminosity as the rate at which a star emits packets (photons) of energy…
 Nick Devereux 2006

16. Flux The further you move away from the star, the flux of photons,
(measured in units of W/m2) passing through a 1m x 1m area
goes down as the reciprocal of the distance squared;
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17. Flux (continued) Quantitatively, the flux f = L/4D2 Units: W/m2
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18. Magnitudes
The magnitude scale dates back to the Greek astronomer Hipparchus (200 BC).
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19.  Nick Devereux 2006

20. Definition of Magnitude
The human eye perceives, as linear, what are actually logarithmic
differences in brightness.
m = -2.5log f + c
m is the apparent magnitude
f is the flux
c is a constant related to the flux of a zero magnitude star
Note the –2.5, the brighter the star (f increases), the more
negative the magnitude (m decreases).
 Nick Devereux 2006

21. Differences in magnitudes are equivalent to ratio’s of fluxes
This obviates the need to know the constant c, or, the zero point,
of the magnitude scale because;
m1 = -2.5log f1 + c
m2 = -2.5log f2 + c
m1 – m2 = 2.5 log f2 / f1
Note the c’s cancelled (c – c = 0)
Also, beware the subscripts are reversed on either side f the
equals sign.
 Nick Devereux 2006

22. Question: A binary star system has one star that is 8 times brighter than the other. What is the magnitude difference between the two stars?
 Nick Devereux 2006

23. Absolute Magnitude
We are unable to tell just by looking at the night sky if one star
is fainter than another because it is intrinsically fainter (ie. lower
luminosity) or just further away.
To realistically compare stars on an equal basis we introduce the
concept of Absolute magnitude (M) which is the magnitude stars
have if they are all placed at the same reference distance of 10 pc.
 Nick Devereux 2006

24. ----------------------------* (1) 10 pc
* (2)
d(pc)
* (1)
10 pc
f1 = L/4(10)2 and f2 = L/4  d2
M – absolute magnitude = -2.5log f1 +c
m – apparent magnitude = -2.5log f2 +c
Then,
M-m = 2.5log f2 / f1
M-m = 2.5log L 4 (10)2 /4  d2 .L
M-m = 2.5log 100/d2
 Nick Devereux 2006

25. Distance Modulus M-m = 5 – 2.5log d2 M-m = 5 – 5log d
So, the absolute magnitude,
M = m + 5 – 5log d
(remember, the distance d must be in pc)
On rearranging,
m - M= 5log d – 5
Where the quantity m – M is known as the distance modulus
 Nick Devereux 2006

26. Trigonometric Parallax
Of course, to calculate the absolute magnitude of a star, we must
know it’s distance. Distances to nearby stars can be found using
Trigonometric parallax.
 Nick Devereux 2006

27. Parallax Angle  Question:
Given the definitions for angular units provided earlier
show that the parallax angle,  measured in arc seconds is equal to the reciprocal of the distance to the star in pc.
So that,
 = 1/d
The nearest star a Centauri at a distance of 1.3 pc has a parallax angle  = 1/1.3 = 0.77’’. All other stars have even smaller parallaxes.
 Nick Devereux 2006