Download 3. Measuring Distances and Magnitudes

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SYLLABUS:
• Part 1: stars and stellar structure (Dr Miao)
• Part 2: measurements (distance brightness, mass)
• Part 3: extrasolar planets (detection, properties)
• Part 4: star and planet formation
• Part 5: radiation, ( the Sun)
• Part 6: telescopes, detectors, instruments
Assessment Methods:
Examination 70%, class tests 15% each (best 2 of 3 marks)
6 assignments.
Class Test 1: week 16
Class Test 2: week 20
Class Test 3: week 24
Homeworks due: data/time on assignment sheets
Recommended Texts:
Carroll & Ostlie, An Introduction to Modern Astrophysics, AddisonWesley, second edition if possible
THESE LECTURE NOTES, posted at the beginning of each week on my
webpage
http://astro.kent/ac/uk/~df/teaching/ph507
[Note: Changes may occur to the syllabus during the year]
Convenor: Dr Dirk Froebrich:
117 Ingram, x7346,[email protected]
Times: please see your time table
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PART 1: Measurements
DISTANCES
Distance: Distance is an easy concept to understand: it is just a length in
some units such as in feet, km, light years, parsecs etc. It has been
excruciatingly difficult to measure astronomical distances until this
century.
Astronomical Unit: 1AU is the distance from the centre of the Sun at
which a particle of negligible mass, in an unperturbed circular orbit, would
have an orbital period of 365.2568983 days (one Gaussian year).
1AU=149.597.870.691±30m
Measured today by radar-distances to nearby planets, asteroids, their
periods around the Sun and applying the 3rd of Keplers Laws: t2/a3=const
(see later in course…)
Defined by IAU in 2012 as 149.597.870.700m exact!
Unfortunately most stars are so far away that it is impossible to directly
measure the distance using the classic technique of triangulation.
Trigonometric parallax: based on triangulation – need three parameters to
fully define any triangle; e.g. two angles and one baseline.
To triangulate to even the closest stars we would need to use a very large
baseline. In fact we do have a long baseline, because every 6 months the
earth is on opposite sides of the Sun. So we can use as a baseline the major
axis of the earth's orbit around the Sun.
BASELINE: 2 x Earth-Sun distance = 2 Astronomical Units (AU)
Examples:
Radius of Earth:
Radius of Sun:
1 AU:
6.372,797 km
695.500
km
149.597.871
km
(1.496 x 10 11 m)
The parallactic displacement of a star on the sky as a result of the Earth’s
orbital motion permits us to determine the distance from the Sun to the star
by the method of trigonometric (heliocentric) parallax. We define the
trigonometric parallax of the star as the half angle p subtended, as seen
from the star, by the Earth’s orbit of radius 1 AU. If the star is at rest with
respect to the Sun, the parallax is half the maximum apparent annual
angular displacement of the star as seen from the Earth.
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1 radian is defined as:
1 radian =
360
 57.3 degrees = 206265 arc seconds.
2
There are 2rad in a circle (360˚), so that 1rad equals 57˚17´44.81” (206,
264.81”).
Therefore, the solar disc subtends an angle of:
2x700.000/150.000.000rad = 0.01rad, i.e. about half a degree.
Independent distance unit is the light year [ly]:
c * t(year) = 9.47 * 10 15m
The light year is not used much by professional astronomers, who work
instead with the unit of similar size called the parsec, where
1parsec = 1pc = 206.265AU = 3.086 x 1016m = 3.26ly.
The measurement and interpretation of stellar parallaxes is a branch of
astrometry, and the work is exacting and time-consuming. Consider that
the nearest stellar system, Proxima/Alpha Centauri AB (Rigil Kentaurus), at
a distance of 1.29pc, has a parallax of only 0.772”; all other stars have
smaller parallaxes!
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Remember diffraction limit for circular apperture: dr[rad] = 1.22 *  / D
 in the optical 1” resolution corresponds to an 11.5cm diameter telescope
tan p=
1 AU d
or d≈
1
AU
p
where p is in rad. For small angles (p<<1):
sin(p)=p –p3/3!+p5/5!... and cos(p)=1-x 2/2!+x4/4!... and hence
tan(p)=sin(p)=p
To convert to arcseconds:
d≈
1
pc
p''
or
d≈
206300
AU
.
p''
Defines the parsec – parallax second abbreviation!
Technique:
The ground-based trigonometric parallax of a star is determined by taking
photographs of a given star field from a number (about 20) of selected
points in the Earth’s orbit at different times of the year. The comparison
stars selected are distant background stars of nearly the same apparent
brightness as the star whose parallax is being measured.
Corrections are made for atmospheric refraction and dispersion and for
detectable motions of the background stars; any motion of the star relative
to the Sun is then extracted. What remains is the smaller annual parallactic
motion; it is recognized because it cycles annually.
Because a seeing resolution of 0.25” is considered exceptional (more
typical it is 1”), it may seem strange that a stellar position can be
determined to ±0.01” in one measurement; this accuracy is possible because
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we are determining the centre of a Gaussian stellar image.
Astrometry: Technological advances (including the Hubble Space
Telescope) have improved parallax accuracy to 0.001” within a few years.
Prior to 1990, less than 10,000 stellar parallaxes had been measured (and
only 500 known well), but there are about 1012 stars in our Galaxy.
Space observations made by the European Space Agency with the
Hipparcos mission (1989-1993) accurately determined the parallaxes of
many more stars. Though a poor orbit limited its usefulness, Hipparcos was
expected to achieve a precision of about 0.002”. It actually achieved 0.001”
for 118,000 stars. The method of trigonometric parallax is important
because it is our only direct distance measurement technique for stars.
Radio interferometry on Masers has been used to determine directly
the distance to the Galactic Center via the parallax method
(d=7.2kpc).
Future:
Launched in December 2013, Gaia will be measuring positions, proper
motions, brightness and parallaxes for objects with a limiting magnitude
V=20mag (1billion stars); accuracy 20µas at V=15mag and 7µas at
V=10mag (hair at 1000km); over a 5yr period.
Plus radial velocities with 500m/s accuracy for 40 million stars
SIM: Space Interferometer Mission from the US – pointed deep
observations down to V=20mag. It will be able to measure parallaxes of
10micro-arcseconds. It consists of a rotating frame holding three
telescopes. Some aims:
- Accurate distances even to the Galactic centre 8000pc away.
Canceled in 2010!
However, the Parallax remains limited to relatively nearby objects. To go
further, we construct the COSMIC DISTANCE LADDER. If we can
estimate the luminosity of a star from other properties, they can be used as
STANDARD CANDLES (L  r-2 … see below)
Example: Sirius A (SpT: A1, V=-1.44mag) has an observed parallax of
0.37921”. Distance: 1/0.37921=2.637pc.
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2 LUMINOSITY
We can actually only measure the radiant flux of a flame and need to make
a few assumptions to find the true luminosity. Luminosity depends on the
distance and extinction (as well as relativistic effects).
The measured flux f is in units of [W/m2], the flow of energy per unit area.
The radiated power L (Luminosity, [W]), ignoring extinction, is given by
an inverse square law:
L
4πd2
L
d 2=
4πf
f=
showing that a standard candle can yield the distance.
The Stellar Magnitude Scale
The first stellar brightness scale - the magnitude scale - was defined by
Hipparchus of Nicaea and refined by Ptolemy almost 2000 years ago. In
this qualitative scheme, naked-eye stars fall into six categories: the
brightest are of first magnitude, and the faintest of sixth magnitude. Note:
The brighter a star, the smaller the value of its magnitude.
Sun: mV=-26.72mag; Sirius: mV=-1.46mag; Deneb: mV=1.25mag
In 1856, N. R. Pogson verified William Herschel’s finding that a firstmagnitude star is 100 times brighter than a sixth-magnitude star and the
scale was quantified. Because an interval of five magnitudes corresponds to
a factor of 100 in brightness, a one-magnitude difference corresponds to
a factor of 1001/5 = 2.512.
This definition reflects the operation of human vision, which converts
equal ratios of actual intensity to equal intervals of perceived intensity. In
other words, the eye is a logarithmic detector.
The magnitude scale has been extended to positive magnitudes larger than
+6.0mag to include faint stars (e.g. the 5m telescope on Mount Palomar can
reach to magnitude +23.5mag) and to negative magnitudes for very bright
objects (the star Sirius is magnitude -1.4). The limiting magnitude of the
Hubble Space Telescope is about +30mag, the planned 40m E-ELT
telescope will reach about +38mag.
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Astronomers find it convenient to work with logarithms to base 10 rather
than with exponents in making the conversions from brightness ratios to
magnitudes and vice versa.
Consider two stars s1 and s2 with a respective apparent brightness m1 and
m2. The ratio of their fluxes f1/f2 corresponds to the magnitude
difference m1–m2. Because a one-magnitude difference means a brightness
ratio of 1001/5= 2.512, (m1 – m2) magnitudes refer to a ratio of (1001/5)(m1-m2) = 100-(m1-m2)/5, or
f1/f2 = 100-(m1-m2)/5
Taking the log10 of both sides and remember:
log(xa) = a*log(x) and log 10a = a log 10 = a,
log (f1/f2) = -[(m1 – m2)/5] log 100 = -0.4(m1 – m2)
or
m1 – m2 = -2.5 log (f1/f2)
This last equation defines the apparent magnitude; note that m1<m2
when f1>f2, that is: brighter objects have numerically smaller
magnitudes!!!
Also note that when the brightness is observed at the Earth, physically they
are fluxes. Apparent magnitude is the astronomically peculiar way of
talking about fluxes.
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Here are a few worked examples:
(a) The apparent magnitude of the variable star RR Lyrae ranges from
+7.1mag to +7.8mag – magnitude amplitude of 0.7mag. To find the relative
increase in brightness from minimum to maximum, we use:
log (fmax / fmin) = 0.4 x 0.7 = 0.28
so that
fmax / fmin = 100.28 = 1.91
This star is almost twice as bright at maximum light, than at minimum.
(b) A binary system consists of two stars (a and b), with a brightness ratio
of 2; however, we see them unresolved as a point of +5.0mag. We would
like to find the magnitude of each star. The magnitude difference is
mb - ma = 2.5 log (fa / fb) = 2.5 log 2 = 0.75mag
Since we are dealing with brightness ratios, it is not right to put
ma+mb=+5.0mag. The sum of the fluxes (fa + fb) corresponds to a fifthmagnitude star. Compare this to a 100-fold brighter star, of magnitude
m0=0.0 and flux f0:
ma+b - m0 = 2.5 log [f0 / (fa + fb)]
or
5.0 - 0.0 = 2.5 log 100 = 5.
But
fa = 2 fb, so that fb = (fa + fb)/3.
Therefore:
(mb-m0) = 2.5 log (f0/fb) = 2.5 log 300 = 2.5x2.477 = 6.19mag.
The magnitude of the fainter star is 6.19mag, and from our earlier result on
the magnitude difference, that of the brighter star is 5.44mag.
What units are used in astronomical photometry?
The well-known magnitude scale of course, which has been calibrated
using standard stars which (hopefully) do not vary in brightness.
But how does the astronomical magnitude scale relate to other photometric
units? Here we assume V (visual) magnitudes, unless otherwise noted,
which are at least approximately convertible to lumen (lm), candela (cd),
and lux (lx). Candela is the SI base unit.
1cd is the luminous intensity emitted in a particular direction of a 555nm
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monochromatic light with 1/683W/sr.
1lx = 1lm/m2 = 1cd*sr/m2
non-SI units used sometime: 1phot=1lm/cm 2
An mv=0 star outside Earth's atmosphere is 2.54 * 10-6 lux
= 2.54 * 10-10 phot
Luminance: (1 nit = 1 candela per square meter
1 stilb = 1 candela per square centimeter)
One mv= 0 star per square degree outside Earth's atmosphere
= 0.84 * 10-2 nit
= 8.4 * 10-7 stilb
One mv = 0 star per square degree inside clear unit airmass
= 0.69 * 10-2 nit
= 6.9 * 10-7 stilb
One star, Mv=0 outside Earth's atmosphere = 2.45* 1029 cd.
3 Attenuation in the earth’s atmosphere
Light incident on the Earth's atmosphere from an extraterrestrial source is
diminished by passage through the Earth's atmosphere. Thus, sources will
always appear less bright below the Earth's atmosphere than above it.
1 clear unit airmass transmits 82% in the visual, i.e. it dims by 0.215mag.
Air Mass is the path length that light from a celestial object takes through
Earth’s atmosphere relative to the length at the zenith. The airmass is 1 at
the zenith and roughly 2 at an altitude of 30°. For small zenith angles (z) it
can be calculated simply as the secant of z.
A = sec(z) = 1 / cos(z)
In a curved atmosphere the airmass is usually smaller than 40, and the
above formula does not apply. A good approximation is (z in degrees)
A = 1 / (cos(z) + 0.50572*(96.07995-z)^-1.6364)
Note that a unit airmass at Mauna Kea (with a mean barometric pressure of
605 millibar – or at about 4500m above sea level) is equivalent to 0.60
airmass at sea level.
From a practical standpoint, you can see that, for example:
At z=60deg you look through 2 airmasses (obtainable from plane parallel
approximation).
At z=71deg you look through 3 airmasses.
At the limit of z=90deg you look through 38 airmasses.
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Atmospheric extinction at optical wavelengths is due primarily to two
phenomena:
1 Absorption:
On UV side primary absorption is Ozone (O3)
On IR side water vapor and CO2.
Atmospheric extinction across the electromagnetic spectrum: Figure
from Observational Astronomy by Lena, Lebrun and Mignard.
As may be seen, most EM radiation is blocked by the Earth, and only a few
atmospheric windows allow certain wavelengths through:
▪ Optical
▪ various near and mid-infrared ranges
▪ Millimeter and radio wavelengths
▪ increasing elevations open up new atmospheric windows
▪ On high mountains and the Antarctic plateau (where the atmosphere is
compressed -- equivalent to higher altitude at warmer latitudes) more midand far- infrared wavelengths are accessible.
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Scattering:
Two mechanisms depend on the size (a) of the scattering particle:
Molecular scattering: scattering radius a << wavelength
▪ Mainly Rayleigh scattering (elastic - energy of scattered photons is
preserved), which has a scattering cross-section as function of wavelength.
Aerosol scattering: scattering radius a >~ 1/10 wavelength
▪ Mie scattering is light scattered by "large" [relative to wavelength of light]
spheres, and the strict theory accounts for Maxwell’s equations in the
context of all kinds of reflections (external and internal) and surface waves
on the scattering particle, polarization, etc.
The sky seems more/deeper blue when you look at greater angles from the
Sun because that is mostly Rayleigh scattering, while close to the Sun the
sky appears "whiter" because this is primarily Mie scattering.
▪ Aerosols are highly variable from night to night.
▪ air pollution
▪ dust = "haze"
▪ volcanic ash (can be horrible)
▪ dust storms
Apparent Magnitude
Apparent magnitude is an irradiance or illuminance, i.e. incident flux per
unit area, from all directions. Of course a star is a point light source, and
the incident light is only from one direction.
Apparent magnitude per square degree is a radiance, luminance, intensity,
or "specific intensity". This is sometimes also called "surface brightness".
Still another unit for intensity is magnitudes per square arcsec, which is
the magnitude at which each square arcsec of an extended light source
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shines.
Only visual magnitudes can be converted to photometric units. U, B, R or I
magnitudes are not easily convertible to luxes, lumens etc., because of the
different wavelengths intervals used. The conversion factors would be
strongly dependent on e.g. the temperature of the blackbody radiation or,
more generally, the spectral distribution of the radiation. The conversion
factors between V magnitudes and photometric units are only slightly
dependent on the spectral distribution of the radiation.
Here we're not interested in the photometric response of some detector with
a well-known pass band (e.g. the human eye, or some astronomical
photometer). Instead we want to know the strength of the radiation in
absolute units: watts etc. Thus we have:
Radiance, intensity or specific intensity:
W m-2 ster-1 [Å-1]
SI unit
erg cm-2 s-1 ster-1 [Å -1]
CGS unit
-2 -1
-1
-1
photons cm s ster [Å ] Photon flux, CGS units
(1erg = 10-7 joule)
Irradiance/emittance, or flux:
W m-2 [Å-1]
SI unit
-2 -1
-1
erg cm s [Å ]
CGS unit
-2 -1
-1
-1
photons cm s ster [Å ] Photon flux, CGS units
Note the [Å-1] within brackets. Fluxes and intensities can be total (summed
over all wavelengths) or monochromatic ("per Angstrom Å" or "per
nanometer").
In Radio/Infrared Astronomy, the unit Jansky is often used as a measure of
irradiance at a specific wavelength intervall, and is the radio astronomer's
equivalence to stellar magnitudes.
The Jansky is defined as: 1Jy = 10-26 W m-2 Hz-1.
4. Absolute magnitude (represents a total flux)
Absolute Magnitude and Distance Modulus
So far we have dealt with stars as we see them, that is, their fluxes or
apparent magnitudes, but we want to know the intrinsic luminosity of a
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star. A very luminous star will appear dim if it is far enough away, and a
low-luminosity star may look bright if it is close enough.
Our Sun is a case in point: if it were at the distance of the closest star
(Alpha Centauri), the Sun would appear slightly fainter to us than Alpha
Centauri does on Earth. Hence, distance links fluxes and luminosities.
Sun’s apparent magnitude: mV=–26.83mag
The solar irradiance is the amount of incoming solar radiation per unit area,
measured on the outer surface of the Earth's atmosphere, in a plane
perpendicular to the rays. The solar constant includes all types of solar
radiation, not just the visible light. It is measured by satellite to be roughly
1366Wm-2.
The luminosity of a star relates to its absolute magnitude, which is the
magnitude that would be observed if the star were placed at a distance of
10 pc from the Sun, in the absence of interstellar extinction. By convention, absolute magnitude is capitalized (M) and apparent
magnitude is written lowercase (m). The inverse-square law of radiative
flux links the flux f of a star at a distance d to the flux, F, it would have if
it were at a distance d = 10 pc:
F / f = (d / D) 2 = (d / 10) 2
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If M corresponds to F and m corresponds to the flux f, then
m - M = 2.5 log (F / f ) = 2.5 log (d/10)2 = 5 log (d / 10)
Expanding this expression, we have useful alternative forms. Since
m1 – m2 = 5 log d1 - 5 log d2
defining the absolute magnitude m2 = M at d2 = 10 pc, so m1 = m and d2 =
d,
m - M = 5 log d - 5
M = m + 5 - 5 log d
In terms of the parallax, using d[pc]=1/p[“]
M = m + 5 + 5 log p”
Here d is in parsecs and p” is the parallax angle in arc seconds.
The quantity m - M is called the distance modulus, for it is directly
related to the star’s distance. In many applications, we refer only to the
distance modulus of objects rather than converting back to distances in
parsecs or light-years.
EXAMPLE:
Luminosity – Absolute magnitude
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Absolute magnitudes for stars generally range from -10mag to +17mag. The
absolute magnitude for galaxies can be much lower (brighter). For example,
the giant elliptical galaxy M87 has an absolute magnitude of -22mag. Many
stars visible to the naked eye have an absolute magnitude which is capable
of casting shadows from a distance of 10pc; Rigel (-7.0mag), Deneb (7.2mag), Naos (-6.0mag), and Betelgeuse (-5.6mag).
Magnitudes at Different Wavelengths
The kind of magnitude that we measure depends on how the light is filtered
anywhere along the path of the detector and on the response function of the
detector itself. So the problem comes down to how to define standard
magnitude systems.
Magnitude Systems
Detectors of electromagnetic radiation (such as the photographic plate, the
photoelectric photometer, CCDs or the human eye) are sensitive only over
given wavelength bands. So a given measurement samples but part of the
radiation arriving from a star.
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Multi wavelength Astronomy: Four images of the Sun, made using (a)
visible light, (b) ultraviolet light, (c) X rays, and (d) radio waves. By
studying the similarities and differences among these views of the same
object, important clues to its structure and composition can be found.
Photographic magnitude
Because the flux of starlight varies with wavelength, the magnitude of a
star depends upon the wavelength at which we observe. Originally,
photographic plates were sensitive only to blue light, and the term
photographic magnitude (mpg) still refers to magnitudes centered around
420nm (in the blue region of the spectrum).
Visual magnitude
Similarly, because the human eye is most sensitive to green and yellow,
visual magnitude (mv) or the photographic equivalent photo visual
magnitude (mpv) pertains to the wave-length region around 540nm.
Filters
Today we can measure magnitudes in the infrared, as well as in the
ultraviolet, by using filters in conjunction with the wide spectral sensitivity
of photoelectric photometers.
In general, a photometric system requires a detector, filters, and a
calibration (in energy units). The properties of the filters are typified by
their effective wavelength, 0, and band pass,  which is defined as
the full width at half maximum (FWHM) in the transmission profile.
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The three main filter types are wide ( ≈ 100nm), intermediate (≈
10nm), and narrow (≈1nm) band filters. There is a trade-off for the
bandwidth choice: a smaller  provides more spectral information but
admits less flux into the detector, resulting in longer integration times.
For a given range of the spectrum, the design of the filters makes the
greatest difference in photometric magnitude systems.
A commonly used wide-band magnitude system is the UBV system: a
combination of ultraviolet (U), blue (B), and visual (V) magnitudes,
developed by H. L. Johnson. These three bands are centered at 365, 440,
and 550nm; each wavelength band is roughly 100nm wide. In this system,
apparent magnitudes are denoted by U, B or V and the corresponding
absolute magnitudes are sub-scripted: M U, MB or MV.
To be useful in measuring fluxes, the photometric system must be
calibrated in energy units for each of its band passes. This calibration turns
out to be the hardest part of the job. In general, it relies first on a set of
standard stars that define the magnitudes, for a particular filter set and
detector; that is, these stars define the standard magnitudes for the
photometric system to the precision with which they can be measured.
Infrared Windows
The UBV system has been extended into the red and infrared (in part
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because of the development of new detectors, such as CCDs, sensitive to
this region of the spectrum). The extensions are not as well standardized as
that for the Johnson UBV system, but they tend to include R and I in the
far red and J, H, K, L, and M in the infrared.
As well as measuring the properties of individual stars at different
wavelengths, observing at longer wavelengths, particularly in the infrared,
allows us to probe through clouds of small solid dust particles, as seen
below
A visible-light (left) vs. 2MASS infrared-light (right) view of the central
regions of the Milky Way galaxy graphically illustrating the ability of
infrared light to penetrate the obscuring dust. The field-of-view is 10x10
degrees
Infrared pass bands which allow transmission (low absorption):
J Band: 1.3 microns
H Band: 1.6 microns
K band: 2.2 microns
L band 3.4 microns
M band 5 microns
N band 10.2 microns
Q band 21 microns
Bolometric magnitudes (the magnitude of a star across all wavelengths)
can be converted to total radiant energy flux: A star of Mbol = 0mag
radiates 2.97 * 1028W.
System is defined by Vega at 7.76pc from the Sun with an apparent
magnitude defined as zero.
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With Lbol = 50.1LSUN and Mbol = 0.58mag.
Sun: mbol = -26.8mag
Full moon: -12.6mag
Venus (at brightest): -4.4mag
Sirius: -1.55mag
Brightest quasar: 12.8mag
For Vega: mb = mv = 0.0mag
mk = +0.02mag
Sun: Mb = 5.48mag, Mv = 4.83mag, Mk = 3.28mag
Colour Index/Colour:
B-V, J-H, H-K are differences in magnitude, i.e. they are flux ratios.
Cooler, redder objects possess higher values. It is independent of the
star’s distance.
Extinction
Interstellar Medium modifies the radiation. Dust particles with size of order
of the wavelength of the radiation. Blue radiation is strongly scattered
compared to red: blue reflection nebulae and reddened stars. Hence a colour
excess is produced.
Colour Excess:
E (B-V) = (B-V) - (B-V)o
This measures the reddening.
Modified distance modulus:
m() = M() + 5 log d – 5 + A()
where A() is the extinction due to both scattering and absorption, along
the entire line of sight. It is strongly wavelength dependent. The optical
depth (is given by
I
exp(−τ )= .
Io
Therefore: A() = 1.086 (show this as homework)
The optical depth is:  * N
where N is the total column density of dust (m-2) between the star and the
observer, and  is the scattering or absorption cross-section (m2).
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In the Interstellar Medium (ISM) an empirical Law relates extinction to
reddening:
AV / E(B-V) = RV = 3.1 + - 0.2
Thus, an objects emission is modified/attenuated by the ISM and the
Earth’s atmosphere. But these effects can be estimated.
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The Hertzsprung-Russell Diagram
In 1911, Ejnar Hertzsprung plotted the first such two-dimensional diagram
(absolute magnitude versus spectral type) for observed stars, followed
(independently) in 1913 by Henry Norris Russell.
The simple HR diagram represents one of the great observational syntheses
in astrophysics. Note that any two of luminosity, magnitude, temperature,
and radius could be used, but visual magnitude and temperature are
universally obtained quantities.
An original idea was that a star was born hot (early type) and cooled (late
type).
It’s a particular colour-magnitude diagram.
Important stars: no obvious pattern…Sirius B, Betelgeuse in opposite
corners:
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Nearby stars: main-sequence appears. Most stars are less luminous and
cooler than the Sun ( Centauri, nearest to us and a triple system, is
similar). Note the hot small stars: the white dwarfs.
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Most stars have properties within the shaded region known as the main
sequence. The points plotted here are for stars lying within about 5 pc of
the Sun. The diagonal lines correspond to constant stellar radius, so that
stellar size can be represented on the same diagram as luminosity and
temperature.
The first H-R diagrams considered stars in the solar neighbourhood and
plotted absolute visual magnitude, M, versus spectral type, which is
equivalent to luminosity versus spectral type or luminosity versus
temperature. Note (a) the well-defined main sequence (class V) with everincreasing numbers of stars toward later spectral types and an absence of
spectral classes earlier than A1 (Sirius), (b) the absence of giants and
supergiants (class III and I), and (c) the few white dwarfs at the lower left.
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The brightest stars:
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An H-R diagram for the 100 brightest stars in the sky. Such a plot is biased
in favour of the most luminous stars--which appear toward the upper rightbecause we can see them more easily than we can the faintest stars. These
are the GIANTS and SUPERGIANTS
In contrast, the H-R diagram for the brightest stars includes a significant
number of giants and supergiants as well as several early-type mainsequence stars. Here we have made a selection that emphasises very
luminous stars at distances far from the Sun. Note that the H-R diagram of
the nearest stars is most representative of those throughout the Galaxy: the
most common stars are low-luminosity spectral type M.
The most prominent feature of the H-R diagram is the Main Sequence:
 Strong correlation between Luminosity and Temperature.
 Hotter stars are Brighter than cooler stars along the M-S.
 About 85% of nearby stars, including the Sun, are on the M-S.
All other stars differ in size:
Giants & Supergiants:
• Very large radius, but same masses as M-S stars
White Dwarfs:

Very compact stars: ~R earth but with ~0.6 Msun!
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Standard Candles
1 Spectroscopic Parallax
The term spectroscopic parallax is a misnomer as it actually has nothing to
do with parallax.
Hertzsprung and Russell deduced the main-sequence for nearby stars,
relating their luminosity to their colour.
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Groups of distant stars should also lie along the same main-sequence strip.
However they appear very dim, of course due to their distance. On
comparison of fluxes, we determine the distance. This works out to about
100,000 pc, beyond which main-sequence stars are too dim.
If we take a spectrum of a star we can determine: its Spectral Class /
Spectral Type.
Using photometry we can measure its apparent magnitude, m V.
If we use B and V filters we can also determine the blue apparent
magnitude, B and thus determine: the star's colour index, CI = B - V.
Knowing either the star's spectral type or colour index allows us to place
the star on a vertical line or band along a Hertzsprung - Russell diagram.
If we also know its luminosity class we can further constrain its position
along this line. Hence we can distinguish between a red supergiant, giant or
main sequence star, for example.
Once we know its position on the HR diagram we can infer what its
absolute magnitude, MV should be by either reading off across to the
vertical scale of the HR diagram or looking it up from a reference table. A
main sequence (luminosity class V) star with a colour index of 0.0 (i.e. A0
V) has an absolute visual magnitude of +0.9mag for example.
Now knowing m from measurement and inferring M we can use the
distance modulus equation:
m - M = 5 log(d/10)
to find the distance to the star, d, in parsecs.
EXAMPLE
Gamma Crucis is an M3.5 III star, a red giant.
Measured values: mV = +1.59mag and a B-V colour index of +1.60mag.
Using its spectral type and luminosity class we can place it where the red
circle is on the HR diagram. Reading across to the vertical axis this
corresponds to an absolute magnitude of about -0.8mag.
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….show that the distance is about 30pc………?
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2 Cepheids as Standard Candles: The Period-Luminosity
Relationship
Cepheids are bright (103-104 times the Sun) variable Population I stars,
named after the prototype -Cep. They show an important connection
between their period of variability and luminosity: the pulsation period of a
Cepheid variable is directly related to its median luminosity.
This relationship was first discovered from a study of the variables in the
Magellanic Clouds, two small nearby companion galaxies to our Galaxy
that are visible in the night sky of the southern hemisphere. To a good
approximation, you can consider all stars in each Magellanic Cloud to be at
the same distance. Henrietta Leavitt, working at Harvard in 1912, found
that the brighter the median apparent magnitude (and so the luminosity,
since the stars are the same distance), the longer the period of the Cepheid
variable. A linear relationship was found.
Harlow Shapley recognized the importance of this period-luminosity (PL) relation-ship and attempted to find the zero point, hence the knowledge
of the period of a Cepheid would immediately indicate its luminosity
(absolute magnitude) and thus its distance.
This calibration was difficult to perform because of the relative scarcity of
Cepheids and their large distances. None are sufficiently near to allow a
trigonometric parallax to be determined, so Shapley had to depend upon the
relatively inaccurate method of statistical parallaxes. His zero point was
then used to find the distances to many other galaxies. These distances are
revised as new and accurate data become available. Right now, some 20
stars whose distances are known reasonably well (because they are in open
clusters) serve as the calibrators for the P-L relationship.
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Further work showed that there are two types of Cepheids, each with its
own separate, almost parallel P-L relationship.
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The classical Cepheids are the more luminous, of Population I, and found
in spiral arms. Population II Cepheids, also known as W-Virginis stars
after their prototype, are found in globular clusters and other Population II
systems.
Classical Cepheids have periods ranging from one to 50 days (typically five
to ten days) and range from F6 to K2 in spectral type.
Population II Cepheids vary in period from two to 45 days (typically 12 to
20 days) and range from F2 to G6 in spectral type.
Population I and II Cepheids are both regular / periodic variables; their
change in luminosity with time follows a regular cycle. The empirically
derived relationship between a Type I Cepheid's period P (in days), and its
absolute magnitude MV is given by
Cepheids are among the brightest individual stars. Hence, they can be
used to determine distances to quite far away galaxies (to about 5Mpc).
HST stretched this to 18Mpc (the Virgo cluster).
3 Tully-Fisher Relation
The Tully-Fisher relation, published by astronomers R. Brent Tully and J.
Richard Fisher in 1977, is a standard candle that measures the distance to
rotating spiral galaxies by the width of the galaxy's spectral lines.
The empirically-derived relation states that the luminosity of a galaxy is
directly proportional to the fourth power of its rotational velocity, which
can be calculated from the width of the spectral line, and uses the distance
modulus to find distance from luminosity and apparent magnitude.
In a spiral galaxy, the centripetal force of gas and stars balances the
gravitational force:
m * v2 / R = G * m * M / R2.
If they have the same surface brightness (L/R2 is constant) and the same
mass-to-light ratio (M/L is constant), then L~v4. So, provided we can
measure the velocity v, certain galaxies can be used as standard candles.
E.g. determine v through the 21 cm line of atomic hydrogen in the galaxy.
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Brightest MB = -19mag, rotation velocities 100 - 500km/s
4 Type Ia Supernovae.
The peak output from these supernovae is about M B=-19.33±0.25mag and
almost constant. Therefore, we can infer the distance from the inverse
square law. Being so bright, they act as standard candles to large distances:
out to 1000Mpc.
Why are they standard candles? White dwarfs in binary systems. Material
from a companion red giant is dumped on the white dwarf surface until the
WD reaches a critical mass (Chandrasekhar mass) of 1.4 solar masses.
Explosion occurs with fixed rise and fall of luminosity. The fall in
luminosity is due to radioactive decay of Ni 56, Co56 and Fe56.
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5 Other methods include:
1: time delay of light rays due to gravitational lensing,
2: cluster size influences Compton scattering of CMB radiation and
Bremsstrahlung emission (X-rays). Combining, yields the size estimate
(Sunyaev-Zeldovich effect).
New Method
Reverse argument: knowing the Hubble constant is 72 km/s/Mpc (WMAP
result) distances can be found directly from the redshift!
For nearby galaxies: Distance = velocity / Hubble Constant
Questions
How do we scale the solar system?
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How do we find the distance to gas clouds?
- measure the distance to associated objects, i.e. stars in or close to the
cloud (those objects could be identified e.g. by reflection nebulae). Rather
difficult method, due to high and variable extinction, and/or young age of
many of the stars within dust clouds.
- count the number of stars which are foreground to the cloud. A model of
the distribution of stars within our Galaxy then can lead to statistically
based distances for the cloud.
- Use the galactic rotation curve to determine the distance. This is
ambiguous for clouds closer to the galactic center than the Sun.