Download How astronomers measure brightness and colours – the magnitude

Introduction to Galaxies and Cosmology
VT-13
How astronomers measure brightness and colours
– the magnitude system
Magnitudes
The origin of the techniques astronomers use to measure the brightness of
objects in the sky can be traced back to ancient Greece (120 B.C.), when
the greek philosopher Hipparcos classified stars visible to the naked eye by
magnitudes. The brightest stars were put into the first magnitude class, the
stars were then classified uniformly in magnitudes by their brightness as seen
by the naked eye. The faintest stars were put in the sixth class. Ptolemy
later put this into the first known stellar catalogue, the Almagest, this scale
was used for more than 2000 years and influenced the modern astronomy to
use the magnitude system of measuring luminosities.
The human eye perceives light logarithmically, this corresponds to a
logarithmic relation between the radiation flux (energy per time unit and
area) and magnitude. The magnitude system is a calibrated photometric
system, the definition of the apparent magnitude is:
m1 − m2 = −2.5 log F1 /F2
(1)
Observationally the apparent magnitude is obtained from:
m = −2.5 log F + mzp ,
(2)
where F is the flux observed by the instruments and mzp is the magnitude zeropoint which is found by calibrating the measurements against some source
with known magnitude. Note that a faint object will have a high magnitude,
this is a remainder from Hipparcos magnitude classes.
Photometric filters
There is a wavelength dependence hidden in the magnitude definitions. The
naked eye magnitudes used by astronomers until the arrival of photography,
is in reality not a measurement of the entire flux from an object. The
human eye acts as a photometric filter, picking up photons with energy
corresponding to wavelengths between ∼ 400 nm (blue) and ∼ 800 nm (red).
The expression for the magnitude (equation 2) in reality becomes:
Z λ2
F (λ)Sf (λ)dλ + mzp ,
(3)
mf = −2.5 log
λ1
where λ1 and λ2 are the lower and upper wavelength limits of the filter and
Sf (λ) is the sensitivity curve of the filter (f) under consideration. This equation also describes the wavelength dependence of any filter used in artificial
photometry.
1
Figure 1: Transmission (sensitivity, Sf (λ)) curves of filters UBVRIJKs (from
the ESO instruments WFI and SOFI). The apparent magnitudes in the
different filters covers different parts of the spectrum.
Modern astronomy uses charge-coupled devices (CCD’s) instead of photographic plates to detect photons. In a CCD many pixels (for example
1 024 × 1 024 pixels) form a mosaic of small photodetectors. To study
colours of the objects, magnitudes in different wavelength bands is measured by introducing a filter before detection in the CCD. The sensitivity of
some commonly used filters is shown in figure 1.
In general, filters used in various instruments are not exactly the same.
This means that in order to compare the magnitudes of different measurements it is necessary to correct the magnitudes to the same filter system.
The process of transforming the magnitudes into a specific filter system is
called color correction. A traditional filter system, which has been in use for
many years is the Johnson-Morgan-Cousin system. The magnitudes in the
different filters are given capital letters (central wavelength in parenthesis);
U (365 nm), B (440 nm), V (550 nm), R (650 nm), I (790 nm) or are given
in the subscript (e.g. mV ). The system is then defined by choosing a magnitude zeropoint. The zeropoint is fixed by stating that the star Vega (α
Lyrae) has zero magnitude in all filters. Later on this was revised so that
a standard A0 star would have zero magnitude in all filters, and a mean
calibration has since then been established. The five filters above (U BV RI)
are the optical filters, there are also filters in wavelength intervals outside
the human eye sensitivity (e.g. ultraviolet and infrared filters).
2
Colour indices
The magnitudes can be subtracted from each other to form colour indices
(e.g. B − V ), note that this is equivalent to division of the fluxes in the
two filters. A non-zero colour index means that the apparent flux from the
object is stronger in one filter than in the other. The colour indices are
usually constructed with a redder filter being subtracted from a bluer filter.
If we compare two galaxies which have B − V = 2.0 and B − V = −1.0
we can say that the first galaxy is redder than the second. Note that the
B − V = 0 limit is an arbitrary boundary. If the Johnson-Morgan-Cousin
system is used the reference object is a standard A0 star, this means that
an object with B − V > 0 is redder than the reference, not that it’s red in
an absolute sense.
Colours of stars and galaxies can be used to study the internal properties
of the objects. The colours are also affected by so called extinction, the
interstellar medium between the object and observer will scatter part of the
light and the objects will be observed to be fainter than they are. The
physical process taking place is Rayleigh scattering, this scattering process
has a wavelength dependence where the less energetic (e.g. RI and infrared
filters) radiation is less affected while the shorter wavelength (e.g. ultraviolet
and U B filters) radiation can be completely extinct (to the detection limit
of the observations). Since the “true” colours of main sequence stars are
known (see table 1 or the figure in Exercise session 1), observed colours
of these stars have been used to find the empirical extinction law for the
Galaxy (for the V band):
AV ≈ 3 · E(B − V ),
(4)
where AV is the extinction in magnitudes for the V filter (c.f. equation 6)
and E(B − V ) = (B − V )obs − (B − V )0 is the colour excess for the object.
Intrinsic luminosities and absolute magnitudes
The luminosity of an object is the total amount of energy sent out from the
object per second (the unit is normally Watt) in a wavelength interval (for
an infinite wavelength coverage it’s called bolometric luminosity). The flux
from a source is related to the luminosity through:
F =
L
,
4πd2
(5)
where F is the flux in J/s/m2 , L the luminosity in J/s and d is the distance
to the source in meters.
The absolute magnitude of objects is defined as the apparent magnitude
an object would have if it was at a distance of 10 parsecs (1 parsec =
3.086 × 1016 m = 3.26 light years), quite nearby in astronomical terms. The
3
flux is inversely proportional to the square of the distance, so the absolute
magnitude can be written:
M = m + 5 − 5 log r − A,
(6)
with the distance r in parsecs and A the extinction correction (due to the
presence of interstellar material between the source and observer). Similar
to the apparent magnitudes, the absolute magnitudes for a specific band
are denoted by a capital “M” followed by a subscript for the band (e.g.
MV ). To go from an absolute magnitude for the V band to the bolometric
absolute magnitude the so called bolometric correction (BC) is applied (see
equation 7).
Mbol = MV − BC
(7)
The absolute magnitude gives an estimate of an object’s intrinsic brightness. Similar to equation 1, we can construct the absolute magnitude difference by combining equation 2, 5 and 6. We obtain:
M1 − M2 = −2.5 log L1 /L2 ,
(8)
with M denoting absolute magnitudes and L the luminosities.
Example: How to use the Sun as a reference for luminosity and absolute
magnitude. The Sun has an absolute bolometric magnitude of 4.75 and
a luminosity of 3.83 · 1026 W. The star Vega has an absolute bolometric
magnitude of 0.38. What is the luminosity of Vega?
Answer: We can use equation 8 to solve this problem. Setting M1 to the
bolometric magnitude of the Sun, L1 to the solar luminosity and M2 to the
bolometric magnitude of Vega we can solve the equation to obtain LV ega ,
LV ega = L⊙ · 10
M⊙,bol −MV ega,bol
2.5
4
= 56L⊙
(9)
Table 1: Basic properties for main sequence stars. MV denotes the average
absolute magnitude, BC the average bolometric correction and B − V the
average colour index for the different spectral types of main sequence stars.
Type
O2
O5
B0
B5
A0
A5
F0
F5
G0
The Sun
G5
K0
K5
M0
M5
M8
Radius (R⊙ )
16
14
5.7
3.7
2.3
1.8
1.5
1.2
1.05
1.0
0.98
0.89
0.75
0.64
0.36
0.15
Mass (M⊙ )
158
58
16
5.4
2.6
1.9
1.6
1.35
1.08
1.0
0.95
0.83
0.62
0.47
0.25
0.10
Luminosity (L⊙ )
2,000,000
800,000
16,000
750
63
24
9.0
4.0
1.45
1.0
0.70
0.36
0.18
0.075
0.013
0.0008
R⊙ = 6.96 · 108 m, M⊙ = 1.99 · 1030 kg, L⊙ = 3.83 · 1026 W
5
Temp. (K)
54,000
46,000
29,000
15,200
9,600
8,700
7,200
6,400
6,000
5780
5,500
5,150
4,450
3,850
3,200
2,500
MV
-6.2
-5.6
-4.0
-1.1
0.8
1.8
2.4
3.3
4.2
4.83
4.9
5.9
7.5
8.9
14.5
19.5
BC
4.7
4.0
2.9
1.3
0.3
0.1
0.1
0.1
0.2
0.07
0.2
0.4
0.6
1.2
3.5
6.5
B−V
-0.32
-0.32
-0.30
-0.15
0.0
0.19
0.32
0.41
0.59
0.65
0.69
0.84
1.08
1.41
1.80
>2