Transcript Luminosities and magnitudes of stars

Basic Properties of Stars - 3
 Luminosities
 Fluxes
 Magnitudes
 Absolute magnitudes
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,
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?
Luminosities and magnitudes of stars
d 
da
da
r

2

d

normal n

dA


source



I



r
da
•
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
Luminosities and magnitudes of stars
§3.2
Luminosity is energy passing through closed surface
encompassing the source (units watts)
Luminosity L = IdAdd
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)
Fig 4.3. An energy flux, which, at
distance r from a point source, is
2
distributed over an area A, is
 F falls off as 1 / r
spread over an area 4A at a
distance 2r. Thus, the flux density
 Inverse Square Law
decreases inversely proportional
to the distance squared.
 Solar constant is
1365 w m-2

Brightness, the magnitude scale §4.2-3
 In
120 BC, Greek astronomer, Hipparchus,
ranked stars in terms of importance (ie.
brightness)  “magnitude”
magnitude were brightest  6th magnitude
faintest visible stars (later extended to 0 and -1)
 1st
 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)
Rigel & Betelgeuse - 0th Magnitude Stars
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
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
Brightness and the magnitude scale
1528 Latin translation of Ptolemy’s
Almagest based on Hipparchus of 120 BC
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
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
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.
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?