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## Basic Properties of Stars 3

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### Magnitude scale later standardized so that mag. = 1 is exactly 100 ... Brightness and the magnitude scale ... (a) What is the absolute magnitude M of the Sun? ... – PowerPoint PPT presentation

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Title: Basic Properties of Stars 3

1
Basic Properties of Stars - 3
• Luminosities
• Fluxes
• Magnitudes
• Absolute magnitudes

2
Solid Angle
• The solid angle, ?, that an object subtends at a
point is a measure of how big that object appears
to an observer at that point. For instance, a
small object nearby could subtend the same solid
angle as a large object far away. The solid angle
is proportional to the surface area, S, of a
projection of that object onto a sphere centered
at that point, divided by the square of the
sphere's radius, R. (Symbolically, ? k S/R²,
where k is the proportionality constant.) A solid
angle is related to the surface area of a sphere
in the same way an ordinary angle is related to
the circumference of a circle.If the
proportionality constant is chosen to be 1, the
units of solid angle will be the SI steradian
(abbreviated sr). Thus the solid angle of a
sphere measured at its center is 4? sr,

3
For Fun
• 1) What is the angular size of the Sun as seen
from Earth?
• Radius Sun 7.0 x 105 km
• Distance to Sun 1.5 x 108 km
• 2) What is the solid angle of the Sun as seen
from Earth?
• 3) What fraction of the sky does the disk of the
Sun then cover?

4
Luminosities and magnitudes of stars
r

5
Luminosities and magnitudes of stars
• Consider some source of radiation
• Intensity I? energy emitted at some frequency
?, per unit time dt, per unit area of the source
dA, per unit frequency d?, per unit solid angle
d? in a given direction (?,?) (see p. 151-152)
• Units w m-2 Hz-1 ster-1
• d? da/r2 ? ?d? ?da/r2 4?r2/r2 4?

6
Luminosities and magnitudes of stars 3.2
• Luminosity is energy passing through closed
surface encompassing the source (units watts)
• If source (star) radiates isotropically, its
radiation at distance r is evenly distributed on
a spherical surface of area 4 ? r2
• Flux is then
F L /
4 ? r2 (w m-2)
• F falls off as 1 / r2
• Inverse Square Law
• Solar constant is
1365 w m-2

7
Brightness, the magnitude scale 4.2-3
• In 120 BC, Greek astronomer, Hipparchus, ranked
stars in terms of importance (ie. brightness) ?
magnitude
• 1st magnitude were brightest ? 6th magnitude
faintest visible stars (later extended to 0 and
-1)
• Without realizing it, Hipparchus based his scheme
on the sensitivity of the human eye to flux -
logarithmic scale, not a linear one.
• Perceived brightness ? log (actual flux)

8
Rigel Betelgeuse - 0th Magnitude Stars
9
Brightness and the magnitude scale
• Magnitude scale later standardized so that mag.
1 is exactly 100 x brighter than mag. 6
• Difference of 5 mag factor 100 in brightness
• Difference of 1 mag factor 2.512 in brightness
i.e. (2.512)5 100
• Note smaller mag is brighter star
• We can quantify this definition of magnitude
scale Ratio of two brightness (flux)
measurements is related to the corresponding
magnitudes by b1/b2 100 (m2-m1)/5
• b1 and b2 are fluxes and m1 and m2 are
magnitudes
• NB that it is b1/b2 and m2 - m1

10
Brightness and the magnitude scale
• This is usually expressed in the form
• m2 - m1 2.5 log10 (b1/b2)
• Note that it is m2 - m1 on the left and b1/b2 on
the right
• ratio apparent mag.
difference
brightness  (b1/b2) m2-m1
• 1 100
0
• 10 101
2.5
• 100 102
5.0
• 1000 103
7.5
• 10,000 104
10.0
• 108
20.0

11
Brightness and the magnitude scale

12
1528 Latin translation of Ptolemys
Almagest based on Hipparchus of 120 BC
13
Brightness and the magnitude scale
• This is usually expressed in the form
• m2 - m1 2.5 log10 (b1/b2)
• Note that it is m2 - m1 on the left and b1/b2 on
the right
• ratio apparent mag.
difference
brightness  (b1/b2) m2-m1
• 1 100
0
• 10 101
2.5
• 100 102
5.0
• 1000 103
7.5
• 10,000 104
10.0
• 108
20.0

14
Brightness and the magnitude scale
• Since brightness of a given star depends on its
distance, we define
• Apparent magnitude, m (this represents flux)
magnitude measured from Earth
• Absolute magnitude, M (this represents
luminosity) magnitude that would be measured
from a standard distance of 10 parsecs (chosen
arbitrarily)
• m - M 2.5log10 (B/b)
• Where B is the flux measured at 10 pc and b is
flux measured at distance d to the star

15
Brightness and the magnitude scale
• Using inverse square law, B/b (d/10 pc)2 we
get
• m - M 2.5 log10 (d/10)2 5 log10 (d/10) 5
(log10 d - log10 10 )
• The last term is just 1 so we have
• m - M 5 log10 d - 5 or m - M 5 log10
d/10
• m - M is called the distance modulus and will
appear often.
• d is distance to the star in parsecs.

16
Simple problems
• (a) What is the absolute magnitude M of the Sun?
• (b) How much brighter or fainter in luminosity
is the star Proxima Centauri compared to the Sun?
• Needed data
• msun -26.7 mproxima 11.05
• Parallax of proxima 0.77
• 1 pc 206,265 AU
• (c) Total magnitude of a triple star is 0.0. Two
of its components have magnitudes 1.0 and 2.0.
What is magnitude of the third component?