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PH507
Astrophysics
Professor Glenn White
THE MULTIWAVELENGTH UNIVERSE AND EXOPLANETS
School of Physical Sciences Convenor Prof. Michael Smith
Taught in Term 2
Teaching Provision:
1
PH507
ECTS Credits 7.5
Kent Credits 15 at Level I
30 lectures + 4 workshops + 2 class tests
Prerequisites:
PH300, PH301, PH304
Aims: To provide a basic but rigorous grounding in observational, computational and
theoretical aspects of astrophysics to build on the descriptive course in Part I, and to
consider evidence for the existence of exoplanets in other Solar Systems.
Learning Outcomes:
1. An understanding of the fundamentals of making astronomical observations across the
whole electromagnetic spectrum, including discussion of photometry and spectroscopy,
and the physics of the astrophysical radiation mechanisms.
2. An understanding of the motions of objects in extrasolar systems and the basic techniques
required to solve the 2-body problem to measure their properties.
3. An understanding of observational characteristics of stars, and how their physical
structures are derived from observation and using simple physical models.
4. To be able to discuss coherently the origin and evolution of Solar Systems and be able to
evaluate claims for evidence of Solar Systems other than our own.
SYLLABUS:
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Part 1: measurements
Part 2: radiation
Part 3: dynamics
Part 4: star and planet formation
Part 5: telescopes/instruments
Part 6: stars and stellar structure
Assessment Methods:
Examination 70%, Homework 10%, 1st class test 10%, 2nd class test 10%.
Recommended Texts:
Carroll & Ostlie, An Introduction to Modern Astrophysics, Addison-Wesley, [QB461]
Stuart Clark. Extrasolar Planets, Wiley Press
C.R.Kitchin. Astrophysical Techniques, Adam Hilger Press.
[Note: Changes may occur to the syllabus during the year]
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Prof. Michael Smith:
101 Ingram, x7654, [email protected]
Office hours: 10-12am Wed
Bad weather
Numbers, names
Locations, times of lectures
Lecturers
PH507
Astrophysics
Professor Glenn White
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PART 1: Measurement
LECTURE 1
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 excrutiatingly
difficult to measure astronomical distances until this century.
Unfortunately most stars are so far away that it is impossible to directly
measure the distance using the classic technique of triangulation.
Trignometric 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)
(The average distance from the earth to the sun is called the Astronomical Unit.)
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 angle  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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Astrophysics
Professor Glenn White
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Astrophysics
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Astrophysics
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1 radian is defined as:
360
 57.3 degrees = 206265 arc seconds, approximately. There are 2
2
rad in a circle (360˚), so that 1 radian equals 57˚17´44.81” (206, 264.81”).
1 radian =
Independent distance unit is the light year:
c  t ( year )  9.47 1015 m
The light year is not used much by professional astronomers, who work instead
with a unit of similar size called the parsec, where 1 parsec = 1 pc = 206265
AU = 3.086 x 1016 m = 3.26 light years.

The measurement and interpretation of stellar parallaxes are a branch of
astrometry, and the work is exacting and time-consuming. Consider that
the nearest star, Proxima/Alpha Centauri (Rigil Kent), at a distance of 1.3
pc, has a parallax of only 0.76”; all other stars have smaller parallaxes.
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Astrophysics
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Formula:
tan p 
1AU
d
or
d 
1
AU
p
where p is in radians
for small angles.
To convert to arcseconds:
2.063 105
d 
AU
p ''
or
d 
1
pc .
p"
Technological advances (including the Hubble Space Telescope) have improved
parallax accuracy to 0.001” within a few years. Before 1990, fewer 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 technique for stars.
The ground-based trigonometric parallax of a star is determined by
photographing a given star field from a number (about 20) of selected points in
the Earth’s orbit. 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 recognised 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 we are determining the
centre of the fuzzy stellar image.
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Astrophysics
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In 2011 – 2013, Gaia will be set into orbit with a Soyuz rocket (and SIM Space
Interferometric Mission from the US). It will be able to measure parallaxes
of 10 micro-arcseconds. It consists of a rotating frame holding three
telescopes. Some aims:
…….Accurate distances even to the Galactic centre, 8000 parsecs away.
……..Photometry: accurate magnitudes.
……..Planet quest
……..Reference frame from distant quasars (3C273 is 800 Mpc away)
In the meantime, to go further, we construct the COSMIC LADDER.
If we can estimate the luminosity of a star from other properties, they can be
used as STANDARD CANDLES.
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, ignoring
extinction, is given by:
f 
d2 
L
4d 2
L
4f
’
showing that a standard candle can yield the distance.
The Stellar Magnitude Scale
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Astrophysics
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The first stellar brightness scale - the magnitude scale - was
defined by Hipparchus of Nicea and refined by Ptolemy
almost 2000 years ago. In this qualitative scheme, nakedeye stars fall into six categories: the brightest are of first
magnitude, and the faintest of sixth magnitude. Note that the
brighter the star, the smaller the value of the magnitude. In 1856,
N. R. Pogson verified William Herschel’s finding that a
first-magnitude star is 100 times brighter than a sixthmagnitude 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 +6.0 to include faint stars (the 5-m telescope on
Mount Palomar can reach to magnitude +23.5) 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 +30.
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
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Astrophysics
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versa. Consider two stars of magnitude m and n with
respective apparent brightnesses (fluxes) lm and ln. The
ratio of their fluxes fn / fm corresponds to the magnitude
difference m - n. Because a one-magnitude difference
means a brightness ratio of 1001/5, (m - n) magnitudes refer
to a ratio of (1001/5)m-n = 100(m-n)/5, or
fn / fm = 100(m-n)/5
Taking the log10 of both sides (because log xa = a log x and
log 10a = a log 10 = a),
log (fn / fm) = [(m - n)/5] log 100 = 0.4(m - n)
or
m - n = 2.5 log (fn / fm)
This last equation defines the apparent magnitude; note
that m > n when fn > fm, that is: brighter objects have
numerically smaller magnitudes. Also note that when the
brightnesses are those observed at the Earth, physically they
are fluxes. Apparent magnitude is the astronomically
peculiar way of talking about fluxes.
Here are a few worked examples:
(a) The apparent magnitude of the variable star RR Lyrae
ranges from 7.1 to 7.8 - a magnitude amplitude of 0.7. To
find the relative increase in brightness from mini-mum to
maximum, we use
log (fmax / fmin) = 0.4 x 0.7 = 0.28
so that
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Astrophysics
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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 magnitude +5.0. 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 = 0.75
Since we are dealing with brightness ratios, it is not right to
put ma + mb = +5.0. The sum of the luminosities (fa + fb)
corresponds to a fifth-magnitude star. Compare this to a
100-fold brighter star, of magnitude 0.0 and luminosity l0:
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Astrophysics
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ma+ b - m0 = 2.5 log [l0 / (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 = 2.5 x 2.477 =
6.19.
The magnitude of the fainter star is 6.19, and from our
earlier result on the magnitude difference, that of the
brighter star is 5.44.
What units are used in astronomical photometry?
The well-known magnitude scale of course, which has been calibrated us
stars which (hopefully) do not vary in brightness.
But how does the astronomical magnitude scale relate to other photomet
assume V magnitudes, unless otherwise noted, which are at least approx
convertible to lumes, candelas, and lux'es.
1 mv=0 star outside Earth's atmosphere = 2.54 10-6 lux = 2.54 1010 phot
Luminance: ( 1 nit =1 candela per square metre)
1 mv=0 star per sq degree outside Earth's atmosphere = 0.84E-2
nit
= 8.4 10-7 stilb
1 mv=0 star per sq degree inside clear unit airmass
nit
= 6.9 10-7 stilb
= 0.69E-2
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Astrophysics
Professor Glenn White
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(1 clear unit airmass transmits 82% in the visual, i.e. it dims 0.2
magnitudes)
One star, Mv=0 outside Earth's atmosphere = 2.451029 cd
Apparent magnitude is thus 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 shines.
Only visual magnitudes can be converted to
photometric units. U, B, R or I magnitudes
are not easily convertible to luxes, lumens
and friends, 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.
What units are used in radiometry/infrared astronomy?
Here we're not interested in the photometric response of
some detector with a well-known passband (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:
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Radiance, intensity or specific intensity:
W m-2 ster-1 [Å-1]
SI unit
-2
-1
-1
-1
erg cm s ster [Å ]
CGS unit
photons cm-2 s-1 ster-1 [Å-1] Photon flux, CGS units
Irradiance/emittance, or flux:
W m-2 [Å-1]
SI unit
-1
erg cm-2 s-1 [Å ]
CGS unit
-1
photons cm-2 s-1 ster-1 [Å ] Photon flux, CGS units
Note the [A-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, and is
the radio astronomer's equivalence to stellar magnitudes.
The Jansky is defined as: 1 Jansky = 10-26 W m-2 Hz-1
Absolute magnitude represents a total flux, expressed in
e.g. candela, or lumens.
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
luminosity of a 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.
Hence, distance links fluxes and
luminosities.
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Astrophysics
Professor Glenn White
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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. (Note that
absolute magnitude is the way of talking about luminosity
peculiar to astronomy). By convention, absolute magnitude
is capitalised (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 luminosity F it
would have it if were at a distance D = 10 pc:
F / f = (d / D)2 = (d / 10) 2.
If M corresponds to L and m corresponds to luminosity l,
then
m - M = 2.5 log (F / f ) = 2.5 log (d/10)2 = 5 log (d / 10)
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Astrophysics
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Expanding this expression, we have useful alternative
forms: since
m1  m2  2.5 log
d1
 5 log d1  5 log d2 ,
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,
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 moduli of different objects
rather than converting back to distances in parsecs or lightyears.
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Astrophysics
Professor Glenn White
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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 that
problem comes down to how to define standard magnitude
systems.
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Astrophysics
Professor Glenn White
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Magnitude Systems
Detectors of electromagnetic radiation (such as the
photographic plate, the photoelectric photometer, and the
human eye) are sensitive only over given wavelength
bands. So a given measurement samples but part of the
radiation arriving from a star.
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.
Because the flux of starlight varies with wavelength, the
magnitude of a star depends upon the wavelength at which
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Astrophysics
Professor Glenn White
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we observe. Originally, photographic plates were sensitive
only to blue light, and the term photographic magnitude
(mpg) still refers to magnitudes centred around 420 nm (in
the blue region of the spectrum). 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 540 nm.
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. So
systems
of
many
different magnitudes (colour
combinations) are possible. 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 bandpass, ∆ which is
defined as the full width at half maximum in the
transmission profile. The three main filter types are wide
(∆≈ 100 nm), intermediate (∆≈ 10 nm), and narrow
(∆≈1 nm). 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 centred at 365, 440, and 550 nm; each
wavelength band is roughly 100 nm wide. In this system,
apparent magnitudes are denoted by B or V and the
corresponding absolute magnitudes are sub-scripted: MB or
MV.
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Astrophysics
Professor Glenn White
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To be useful in measuring fluxes, the photometric system
must be calibrated in energy units for each of its
bandpasses. 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 because of the development of new
detectors, such as CCDs, sensitive to this region of the
spectrum). The extensions are not as well standardised 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 loner wavelengths,
particularly in the infrared, allows us to probe through
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Astrophysics
Professor Glenn White
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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 passbands 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 can be converted to total radiant
energy flux: One star of Mbol = 0 radiates 2.97 1028 Watts.
System is defined by Vega at 7.76 parsecs from the Sun
with an apparent magnitude defined as zero.
With Lbol = 50.1 Lsolar and Mbol = 0.58.
Sun: mbol = -26.8
Full moon: -12.6
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Astrophysics
Professor Glenn White
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Venus: -4.4
Sirius: -1.55
Brightest quasar: 12.8
For Vega: mb = mv = 0. mk = +0.02
Sun: Mb = 5.48, Mv = 4.83, Mk = 3.28
Colour Index: B-V, J-H, H-K are differences
magnitude….flux ratios.
But cooler, redder objects possess higher values.
in
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.
Colour Excess:
measures the reddening.
E(B-V) = B-V - (B-V)o
Modified distance modulus:
m() = M() + 5 log d – 5 + A()
where A () is the extinction due to both scattering and
absorption, strongly wavelength dependent. The optical
depth is given by
I
exp(  ) 
.
Io
Therefore A() = 1.086 

The optical depth is

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Astrophysics
Professor Glenn White
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where N is the total column density of dust (m-2) between
the star and the observer and is the scattering/absorption
cross-section (m2).
ISM Law related extinction to reddening:
Av / E(B-V) = 3.2 + - 0.2
Spectroscopic Parallax
Hertzsprung-Russell deduced the main-sequence stars for
nearby objects, relating their luminosity to their colour.
Groups of distant stars should also\line 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.
Cepheids as Standard Candles: The Period-Luminosity
Relationship
Cepheids show an important connection between period
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.
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Astrophysics
Professor Glenn White
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Harlow Shapley recognised the importance of this periodluminosity (P-L) relation-ship and attempted to find the
zero point, for then a knowledge of the period of Cepheid
would immediately indicate its luminosity (absolute
magnitude).
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.
Further work showed that there are two types of Cepheids,
each with its own separate, almost parallel P-L relationship.
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Astrophysics
Professor Glenn White
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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 class.
Population II Cepheids vary in period from two to 45 days
(typically 12 to 20 days) and range from F2 to G6 in spectral
class.
Population I and II Cepheids are both regular, or periodic,
variables; their change in luminosity with time follows a
regular cycle.
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Astrophysics
Professor Glenn White
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Cepheids are bright and distinct. They can be used to
determine distances to quite distant galaxies, to about 5
Mpc. HST stretched this to 18 Mpc (Virgo cluster).
Tully-Fisher Relation
In a spiral galaxy, the centripetal force of gas and stars
balances the gravitational force:
mV2/R = GmM/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 V, certain galaxies can
be used as standard candles. (determine V through the 21
cm line of atomic hydrogen in the galaxy).
Type 1a Supernovae.
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Astrophysics
Professor Glenn White
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The peak light output from these supernovae is always
about Mb = -19.33 +- 0.25. Therefore we can infer the
distance from the inverse square law. Being so bright , they
act as standard cadles to large distances: to 1000 Mpc.
Why are they standard candles? White dwarfs I 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.
Other methods: time delay of light rays due to gravitational
lensing, cluster size influences Compton scattering of CMB
radiation
and
bremsstrahlung
emission
(X-rays).
Combining, yields the size estimate (Sunyaev-Zeldovich
effect). Or, rotational properties of stars with starspots…….
New Method?
Reverse argument: knowing the Hubble constant is 72
km/s/Mpc, (WMAP result), distances can be found directly
from the redshift!
PH507
Astrophysics
Professor Glenn White
Questions
How do we scale the solar system?
How do we find the distance to gas clouds?
PLANET REVIEW
The Terrestrial Solar System
27
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Astrophysics
Professor Glenn White
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In the picture above we see the positions of the asteroid belt
(green) and other near-earth objects
The material in the plane of the Solar System is known as the
Kuiper Belt. Surrounding this is a much larger region known as
the Oort Cloud, that contains material that occasionally falls in,
under the influence of gravity, towards the Sun as comets.
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Astrophysics
Professor Glenn White
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The Sun
At over 1.4 million kilometers (869,919 miles) wide, the Sun
contains 99.86 percent of the mass of the entire solar
system: well over a million Earths could fit inside its bulk.
The total energy radiated by the Sun averages 383 billion
trillion kilowatts, the equivalent of the energy generated
by 100 billion tons of TNT exploding each and every
second.
Planetary configurations
• Some of the definitions below make the assumption of coplanar
circular orbits. True planetary orbits are ellipses with low
eccentricity and inclinations are small so the concepts are
applicable in real cases.
• Copernicus correctly stated that the farther a planet lies from
the Sun, the slower it moves around the Sun. When the Earth
and another planet pass each other on the same side of the Sun,
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Astrophysics
Professor Glenn White
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the apparent retrograde loop occurs from the relative motions of
the other planet and the Earth.
 As we view the planet from the moving Earth, our line of
sight reverses its angular motion twice, and the three-dimensional aspect of the loop comes about because the orbits of
the two planets are not coplanar. This passing situation is
the same for inferior or superior planets.
PH507
Astrophysics
Professor Glenn White
31
 A Retrograde loop occurs when a superior planet moves
through opposition, and occurs as the earth's motion about
its orbit causes it to overtake the slower moving superior
planet. Thus close to opposition, the planet's motion relative
to fixed background stars, follows a small loop.
PH507
Astrophysics
Professor Glenn White
32
PH507
Astrophysics
Professor Glenn White
33