ASTR2100

1
3. The Continuous Spectrum of Light Goals 1.
Discover how distances and luminosities are
established for stars. 2. Describe the
characteristics of light rays. 3. Describe the
characteristics of black body radiation. 4. Learn
how filtered observations of stars can be used to
establish their reddenings and distances.
2
Parallax Parallax, denoted as p, is the defined
as the angle subtended by 1 Astronomical Unit,
A.U., at the distance of a star. In practice one
can observe the annual displacement of a star
resulting from Earths orbit about the Sun as 2p.
Since all stars should exhibit parallax, measured
values (trigonometric parallaxes) are of two
types prel relative parallax, is the annual
displacement of a star measured relative to its
nearby companions pabs absolute parallax, is
the true parallax of a star, or what is measured
for it In the past, all parallaxes were measured
using long focal length refracting telescopes,
but the situation has changed in the past 20-30
years. Such parallaxes were relative parallaxes,
and were adjusted to absolute via pabs prel
correction
3
How parallax is measured.
4
The complication of the parallactic ellipse. In
practice all parallaxes are measured using
only points near maximum parallax
displacement. The concept of relative parallax
is also illustrated.
5
The definition of parallax and parsec
distance at which one Astronomical Unit (A.U.)
subtends an angle of 1 arcsecond. Note, by
definition 1 pc 206265 A.U.
6
The estimated frequency of an average 11th
magnitude star (upper) and an average 16th
magnitude star (lower) as a function of
distance (visual magnitudes dashed line,
photographic magnitudes solid line). 11th
magnitude stars peak for a distance of 250 pc,
corresponding to a correction factor of 0".004.
16th magnitude stars peak for a distance of
800 pc, corresponding to a correction factor
of 0".00125.
7
The distance to any star or object with a
measured absolute parallax is given by The
relative uncertainty in distance is given
by Typical corrections to absolute are
0".003 to 0".005 for the old refractor
parallaxes, but are more like 0".001 for more
recent reflector parallaxes from the U.S. Naval
Observatory. Space-based parallaxes from the
Hipparcos mission are all absolute parallaxes
they were measured relative to all other stars
observed by the satellite. They have typical
uncertainties of less than 1 mas
(milliarcsecond), i.e. lt0".001, although
systematic errors of order 0".001 or more are
suspected in many cases.
8
Example What is the distance to the star Spica
(a Virginis), which has a measured parallax
according to Hipparcos of pabs 12.44 0.86
mas? Solution. The distance to Spica is given
by the parallax equation, i.e. The
uncertainty is The distance to Spica is
80.39 5.56 parsecs.
9
Magnitude System The magnitude system appears
to originate with the ancient Greek astronomer
Hipparchus (190-120 BC), often considered to be
the greatest ancient astronomical observer. He
grouped the stars into six magnitudes (note the
magic number 6) with 1st magnitude stars being
the brightest and 6th magnitude stars being the
faintest detectable. The human eye perceives
brightness almost in logarithmic fashion, so the
best match to the original Hipparchus scheme has
always tended to be an inverse logarithmic scale,
although others have been tried. Currently the
brightnesses of stars are measured using the
Pogson ratio, and are measured using magnitude
differences where f1 and f2 are the observed
brightnesses of the objects.
10
For a star of luminosity L at a distance r, the
light is dispersed over the surface of a sphere
of area 4pr2 by the time it reaches Earth, for no
intervening absorption, i.e. is the radiant
flux we measure at Earth. Therefore Absolute
magnitude, M, is defined as the apparent
brightness a star would have if it were at a
distance of 10 pc. Therefore or mM 5
log r 5, where mM is referred to as the
distance modulus.
11
Example What is the distance to Spica (a
Virginis), which is a B1 III-IV star with an
apparent visual magnitude of V 0.91, given that
B1 III-IV stars typically have an absolute
magnitude of MV 3.7 (Appendix G of
textbook)? Solution. The distance modulus for
Spica is given by Thus is the
spectroscopic parallax distance to Spica. Note
the similarity of the result with the value of
80.4 pc established from the stars Hipparcos
parallax. Also note that there is no associated
uncertainty, unless we assign an uncertainty to
the spectroscopic absolute magnitude.
12
Example A binary system consists of two stars of
magnitude m 6.00 that cannot be easily resolved
in a telescope. How bright does the system appear
if they are measured together? Solution. In
this case So Thus, b1 b2, both stars are
the same brightness. But So m12 m1
0.75 6.00 0.75 5.25. Two stars of equal
brightness always appear 0m.75 brighter than a
single star of the same type.
13
Example The companion to Polaris (V 2.00) has
V 8.60. How much brighter would Polaris appear
if the companion was accidentally included in the
photometer diaphragm when the stars was measured
at the telescope? Solution. Here Thus So
i.e., V 1.9975, which is insignificantly
brighter.
14
Wave Nature of Light Historically the unique
properties of light were found by Danish
astronomer Ole Roemer (16441710), who noted
discrepancies between predicted times for
eclipses and transits of the Galilean satellites
of Jupiter from Keplers Laws and their actual
observed times. The discrepancy results from
finite but varying light travel times to Earth.
Roemers original estimate for the speed of light
of c ? 2.2 ? 108 m/s differs from current best
estimates of 2.9979 ? 108 m/s. Christian Huygens
(16291695) proposed the concept that light is
composed of waves, with wavelength ? and
frequency ? c/?.
15
Any object comprises a large number of point
sources radiating (usually) in non-coherent
fashion like waves radiating from a source. Light
carries energy radially outward at speed c, such
that, at any instant some chosen phase of the
wave (i.e. peak or trough) is located along a set
of concentric shells (rings in two dimensions)
called wavefronts. At large distances from the
source of radiation the wavefronts are
essentially plane parallel planes separated by ?.
Energy is carried perpendicular to the wavefronts
along paths called light rays.
16
Youngs Double Slit Experiment Consider a plane
parallel beam of light incident on a mask
containing two parallel slits, as shown. Take D
gtgt d and assume ? is small. S1P FP, so S1FP is
a skinny triangle. To a good approximation the
angle FS1S2 is the same as ?, i.e. the distance
S2F d sin ?. For constructive interference (a),
d sin ? n?, n 0,1,2 For destructive
interference (b), d sin ? (n½)?, n 1,2, In
other words, waves interfere constructively when
the path difference is multiple number of light
wavelengths. The total energy in the resulting
fringe pattern equals that of the light passing
through the slits.
17
As an example of the expected fringe spacing,
consider a red line at ? 630 nm (6300Å) with d
0.25 mm and the screen at D ? 3000
mm. By the skinny triangle approximation,
s ? D?1, therefore
18
Light as an Electromagnetic Wave Light is a
form of electromagnetic radiation that travels
at speed v c, where c (e0µ0)½ and µ0
magnetic permeability of a vacuum 4p ? 107
kg m/C2 and e0 electrical permittivity of a
vacuum 8.854 ? 1012 C2/J m for c in
m/s. Light carries energy as a wave with
mutually perpendicular electrical and magnetic
fields, the E and B vectors. The direction and
energy carried by a light wave is described by
the Poynting vector, S, which is directed along
the path of a light ray with a magnitude
equivalent to the amount of energy per unit time
interval passing through a plane perpendicular to
the light path.
19
i.e., and where E0 and B0 are the
amplitudes of the electric and magnetic field
vectors. Definitions Flux total energy per
second at all wavelengths crossing a unit area
perpendicular to the Poynting vector S, in units
of W/m2 or ergs/cm2/s (cgs). Flux is the same as
S. Intensity brightness flux received from a
unit solid angle, in units of W/m2/steradian. Lum
inosity total power at all wavelengths radiated
(or in some defined spectral range).
20
So for the light distributed over a
spheroidal surface centred on a star. That means
that So the flux from a distant star varies
as the inverse square of its distance. What about
intensity? Intensity is defined by where O
solid angle subtended by the source. But O
p?2 for small ?.
21
The solid angle is defined by O A/r2, where A
is the area of the source (the gray area). For
small ?, A ? pl2. Thus, we have O pl2/r2, where
l is the fixed size of the source. So the
intensity is given by which is independent
of distance.
22
Example. The Sun. The solar constant, f?(1 A.U.)
1369 W/m2 at the top of Earths atmosphere. 1
A.U. 1.4960 ? 1011 m. Therefore the Suns
bolometric luminosity. The intensity of sunlight
is solved using the radius of 16' subtended by
the Sun, so
23
Since f? and O? both decrease as 1/r2, the
intensity of sunlight is independent of distance,
whereas the flux decreases as 1/r2. In other
words, the intensity of sunlight is the same for
Mars, Jupiter, etc., as it is for Earth. Only the
flux varies.
24
Radiation Pressure. Recall that the kinetic
energy is defined for moving particles as KE
½mv2. For particle energy, which is considered
kinetic, we therefore have the momentum.
For light the equivalent is E mc2, so p
E/c mc. The acting force is dp/dt, so consider
light intercepted by an area A inclined by an
angle ? relative to the direction of the light
rays. In time dt electromagnetic radiation of
flux f carries energy dE to the surface,
where dE f A cos ? dt. The resulting force on
A depends upon how much incident radiation is
absorbed.
25
For pure absorption since the force
Frad(abs.) must be parallel to S. For pure
reflection the incident and reflected beams make
equal angles to the normal to the surface. If
reflection is considered as absorption
immediately followed by emission, then the force
exerted by the surface A is doubled over the case
for absorption.
26
The reflected flux is directed along the normal
to the surface, so along the normal to A
rather than along S.
27
Applications. The Eddington Limit. What happens
when Frad(abs.) ? Fg for a star? Take A 1 m2, ?
0, and assume that the stars atmosphere
absorbs a fraction k of the incident radiation (k
lt 1). So That gives a simple relation
for computing the Eddington luminosity for any
star
28
For M gtgt 1 M?, the mass-luminosity
relation is given by L/L? (M/M?)3.5 for
main-sequence stars. The upper mass limit occurs
for L LED, i.e.
29
But L? 3.851 ? 1026 W. So For k 1/30,
MED 69 M?. (close to optimum value for ?) For
k 1/50, MED 84 M?. For k 1/100, MED 111
M?. For k 1/200, MED 146 M?. For k
1/1000, MED 279 M?. Typical upper limits for
the mass of high-mass main-sequence stars are
70?100 M? observationally, which must correspond
to the Eddington limit for stars of extremely
high radiation fields.
30
Curvature of Comet Dust Tails. The absorption of
sunlight by small grains pushes them radially
away from the Sun. The dust grains assume larger
orbits with correspondingly larger semi-major
axes and, by the vis-viva equation, slower
speeds. They therefore fall behind the comet
nucleus in its orbital motion, producing curved
dust tails. Qualitatively, radiation pressure on
grains effectively reduces the effects of the
Suns gravitational force on them, thereby
producing an effectively smaller solar mass. By
the vis-viva equation Both M and 2/r become
smaller, whereas 1/a is relatively unchanged
(because a is large for comets). Thus, v is
smaller and the dust grains lag behind the comet
nucleus. Or, to conserve total energy, the grains
must lose kinetic energy as they are pushed up
the Suns potential well.
31
Black Body Radiation. One of the great
discoveries of the 19th century was the
characterization of radiation from hot objects.
The major effort of experimentation was to study
the radiation originating from a black body, an
ideal emitter that absorbs all of the light
energy incident upon it and reradiates it. The
experimental construct was a small black box
(perfect absorber) held in thermodynamic
equilibrium, so that every absorption or emission
was balanced by the inverse process and all ways
of measuring the system temperature yield the
same results.
32
Because of the condition of thermodynamic
equilibrium TE, every energy state of the
system must have the same energy kT, where T
absolute temperature (K) and k is the Boltzmann
constant 1.3807 ? 10?23 J/K. The gas pressure
is therefore given by Pg nkT, where n is the
number of gas particles. According to such
ideas, every light frequency between ? 0 and ?
8 must have the same energy. That means that
the total light energy must be infinite, a
contradiction referred to as the ultraviolet (UV)
catastrophe. Max Plancks idea was to suppose
that each light frequency carries a minimum
energy given by E? (minimum) h?, where h
6.6261 ? 10?34 J s is Plancks constant. Thus,
for large frequencies, ?, the value of E?
(minimum) is so large that no light of such
frequencies can be present, i.e. no UV
catastrophe.
33
Black body radiation from an element of surface
area dA.
34
By 1900 Planck was able to match an empirical
formula to observations of black body spectra,
namely where a and b are constants. With
Plancks idea of energy quantization of photons,
the true form of the spectrum could be predicted,
and was found to be described exactly by for
equal units of wavelength, ?, or for equal
units of frequency, ?.
35
Results (i) At any ? the term ehc/?kT ? 1 as T
? 8, so there is a close correlation between B?
and T at any wavelength (Planck curves for
different Ts do not intersect). (ii) The
wavelength of maximum black body flux B?(T)max
correlates with temperature T according to Wiens
Law i.e., for ?max in cm. Example. Where
does a black body with T T? 5779 K reach its
peak output of radiation? By Wiens Law,
i.e., at 5020 Å, in the yellow region of the
spectrum.
36
Example. Where does a black body with T 2900 K
(corresponding to the coolest M-type stars) reach
its peak output of radiation? By Wiens Law,
i.e., at 10,000 Å 1 µm, in the near
infrared region of the spectrum.
37
Photometry. Photometry is the basis for most
studies in astronomy since the resulting data can
be tied directly to information about the
temperatures and intrinsic properties of
astrophysical objects. When considered as a
particle phenomenon, light rays carry energy that
is quantized in multiples of the Planck constant,
i.e. The flux of energy between wavelengths ?
and ?d? is therefore expressed as where n is
the photons /unit area /s /wavelength interval
at wavelength between ?1 and ?2, i.e.
38
Example. The absolute calibration of the UBV
system. For an A0 star (defined such that BV
UB 0.00) with V 10.0, we find that at
the top of Earths atmosphere. Alternate
versions D. F. Gray 2nd
source The corresponding photons arriving at
the top of the atmosphere is therefore where
550 nm 5.5 ? 105 cm. Note stellar
atmospheres specialist prefer cgs units.
39
So of which 80 to 90 reach Earths
surface on a clear night.
40
Astronomical Photometry Systems. Photometry
refers to precision measurements of light energy.
Since stellar continua contain information about
temperature, measurements of stellar flux at two
different wavelengths, ?1 and ?2, provide direct
information about T. For example which can
be solved for T once ?1 and ?2, B?1(T), and
B?2(T) are established. But there are other
factors affecting the ratio (i) stellar
atmospheric gas pressure, which modifies the
ionization of atoms and ions, and (ii) chemical
composition, which affects which atoms/ions are
present in the stars atmosphere.
41
Features of a useful photometric system
include (i) Specific filter/detector
combinations to isolate certain wavelength
regions, and (ii) Accurate apparent magnitudes
for a set of photometric standard stars
(non-variable), using the filter/detector
combinations.
42
UBV System. The UBV system was designed in the
early 1950s by Harold Johnson and Bill Morgan
using a RCA 1P21 photomultiplier, specific
Corning filters, reflecting telescope optics,
observations at 7000 ft altitude, etc.
Filter ?eff(nm) ??(nm) U 365 68 B 440 98 V
550 89 Each passband can be represented by a
sensitivity function S(?) defining the fraction
of incident light energy recorded by the system,
since
43
So Use the Pogson equation to establish the
apparent U magnitude of a program star
unknown relative to a standard star,
e.g. where 1 program star and 2 standard
of known magnitude. For the program star itself
we have Observations involving photometry
therefore consist of using the telescope and
equipment to establish the corrections to the
standard system for each observing session, or
for a particular observing period.
44
In practice, one uses some 10-20 photometric
standards to calibrate ones data, with the
separate data for the standards averaged together
for greater precision and accuracy. The UBV
system is defined by a large set of standards, of
which 10 primary standards are the fundamental
reference points. The UBV system has also been
defined historically using Vega as a reference
object, with V B U for Vega. Actually, V
0.03, BV 0.00, and UB 0.01 for Vega, as
presently calibrated on the Johnson system.
45
Example. On April 7/8, 2003, the Optec SSP-3
photometer at the BGO gave 43,130 counts during a
10s integration when pointed at Polaris through
the V filter, and 716 counts during a 10s
integration when pointed at a nearby comparison
star with V 6.47, as normalized to the same
gain setting and air mass, with sky subtraction
already included. How bright was Polaris? From
the Pogson ratio So, VPolaris Vstd
4.45 6.47 4.45 2.02.
46
Bolometric Magnitudes and Corrections. Stellar
models predict that the integral is
independent of any photometric system. It is
often convenient to correct V magnitudes to the
corresponding total flux from an object, i.e.
over all wavelengths. Such magnitudes are called
bolometric magnitudes, mbol. Bolometric
corrections to visual magnitudes are always in
the sense where and Cbol is chosen by
consensus so that for most normal stars BC lt 0,
i.e. stars put out more light over all
wavelengths than they do in any particular filter
band.
47
As defined The BC is smallest when
is close to 1. We would expect the denominator to
be smaller than the numerator of the fraction,
but the ratio tends to be nearly 1 for stars in
which the black body flux peaks near 550 nm, the
V band. Such stars are of spectral type F0.
48
Colour Indices. Since the ratio of intensities
for a black body measured at two different
wavelengths yields information about the
temperature of the black body, colour indices are
a convenient means of inferring information about
the temperatures of stars. Typically Colour
Index CI m (short ?) ? m (long ?), e.g.,
B?V, U?B, J?H, b?y, etc. In the UBV system
49
Applications of the UBV System. Interstellar
Reddening. In the presence of interstellar
reddening both magnitudes and colours are
affected. Magnitudes suffer from an absorption
term, i.e. VAV, while colours become redder,
i.e. B?V (B?V)0 EB?V. Extinction and
reddening are related, for BV magnitudes through
AV R EB?V, where R is the ratio of total to
selective extinction, i.e. In most nearby
regions of the Galaxy R varies from 2.8 to 3.3,
averaging about R 3.1. In some regions it may
be larger, with values of R of 5 or so being
suggested for some anomalous regions, often H
II regions.
50
Some effects or A more useful arrangement of
the terms is where V?MV is the apparent
distance modulus and V0?MV is the true distance
modulus. The relation can be used for some open
star clusters to establish R, if there is a
spread in reddening for cluster stars
(differential reddening). All cluster stars can
be assumed to lie at the same distance, so a plot
of apparent distance modulus V?MV vs. reddening
EB?V (referred to as a variable-extinction
diagram) will have a slope R and a zero-point
V0?MV, the true distance modulus. Reddenings can
be derived from spectral type-(B?V)0 relations or
from UBV two-colour diagrams.
51
Examples of star clusters with uniformly reddened
stars.
52
Examples of star clusters exhibiting small (left)
and large (right) amounts of differential
reddening.
53
The reddening law can be described by with X
the slope and Y the curvature term. Observations
indicate that Y 0.02 0.01, while X may vary
with R.
54
The two-colour diagram from Carroll and Ostlie.
The intrinsic relation differs from the correct
one, but the location of stars of different
spectral types is essentially correct. Note also
where true black bodies would fall.
55
UBV intrinsic colours are presently tied to
models for non-rotating stars (left), but
differential reddening still dominates the
observed colours (right).
56
How the variable-extinction method is supposed to
work (left), and the types of situations that
occur in practice (right)/
57
Examples of variable-extinction analyses for open
clusters.
58
An example of the variable-extinction method
applied to the OB association Mon OB2 using
spectroscopic distance moduli. The line has slope
R 3.2.
59
The zero-age main sequence, as constructed from
overlapping the main sequences for various open
clusters, all tied to Hyades stars using the
moving cluster method.
60
The present-day zero-age main sequence (ZAMS) for
solar metallicity stars.
61
Why it is important to correct for differential
reddening in open clusters removal of random
scatter from the colour-magnitude diagram.
62
Differences between the core and halo regions of
open clusters. Note also the existence of
main-sequence gaps.
63
Typical open cluster colour-magnitude diagrams
corrected for extinction. Note the pre-main
sequence stars in NGC 2264 (right) and the
main-sequence gaps in NGC 1647 (left).
64
Many young clusters are also associated with very
beautiful H II regions.
65
ZAMS fitting can be done by matching a template
ZAMS (right) to the unreddened observations for a
cluster (left). The precision is typically no
worse than 0.05 (2.5).
66
An interesting example of the importance of ZAMS
for open clusters. The luminous and peculiar B2
Oe star P Cygni belongs to an anonymous open
cluster, so its reddening and luminosity can be
found using cluster stars.
67
The reddening (left) and ZAMS fit (right) for the
P Cygni cluster, an example of the usefulness of
cluster studies.
68
Optical Light detectors. The generalized view of
optical light detectors is The
different types of astronomical detectors
are (i) Photographic plates. These consist of
silver bromide crystals (AgBr?) coating a glass
plate that is exposed to light. Photons interact
with the silver ions to neutralize them, leaving
silver atoms that can be deposited in the
emulsion by development, leaving a negative image
of objects (black stars on a white sky
background).
69
(ii) The eye. Electrical nerve impulses react on
exposure to photons of light, resulting in the
process of vision. The eye is a more precise
detector than believed, especially near the
limits of human vision.
70
(iii) Photomultiplier tubes. In this case photons
incident on a photocathode (in vacuum) liberate
photoelectrons that are multiplied in number via
an electron cascade onto additional cathodes. The
resulting electron impulse generates a small
current in proportion to the photons detected.
The resulting light flux is either recorded
directly via a strip chart recorder or a
voltage-to-frequency converter that produces a
photon count.
71
(iv) CCD detectors. Charge-coupled devices are
multi-layer silicon wafers (chips) containing an
array of independent, light-sensitive elements
(picture elements pixels). Exposure to photons
of light frees electrons in the silicon which
migrate towards the nearest potential well,
typically an anode. Most require cooling to
reduce the effects of thermal energy releasing
electrons. They are also used in Optec
photometers to generate a photon current for
detection.
72
The detection characteristics are usually
quantified by the term quantum efficiency and
a spectral response The typical best
results are qmax ? 0.01 for photographic
plates qmax ? 0.10 for photomultipliers qmax ?
0.8 for CCD detectors
73
A plot of the quantum efficiency QE for a CCD
device (top) as well as the human eye (bottom).