Stars and Their Spectra Flashcards
we model stars as…
self gravitating spheres of gas with the inward pull of gravity balanced by the outward gas pressure
the star’s core must be hot enough to
allow nuclear fusion to take place
ie around 10 million kelvin
the internal structure of a star and the way that energy flows through it depends
primarily on mass but also chemical composition
what does mass of star determine
internal structure of star and way energy flows through it
surface temperature
luminosity
convection dominates in
lower mass stars
radiation dominates in
larger mass stars
where do we learn about flux variations with energy or wavelength
spectra
where do we learn about flux variations with time
light curve
we learn everything we know about stars from
spectra
light curves
EM theory
primary source of information about physical nature of stars
their spectra
planck function
everything constant except frequency
wien’s displacement law derived from it
cooler stars peak wavelength
towards infrared
stellar photospheres are good approximations to a blackbody function but in reality….
their spectra are complicated by the presence of spectral lines and edges
edge
jumps in intensity on spectra
spectral sequence
OBAFGKM
stars in our neighbourhood have roughly
the same chemical composition
stars in our neighbourhood have different spectral lines due to
different temperatures
the stellar surface or atmospheric temperature determines
which ions or excited states are present
‘early’ group in the spectral sequence
OBA stars
7500<T<25000+
He, H lines
A stars have weaker H, stronger metals
‘solar type’ group in the spectral sequence
FG stars
5000<T<7500
very weak He, some metals
G stars have strong metal ions and neutrals
‘late’ group in the spectral sequence
KM stars
coolest with 3500-<T<5000
metal ions, neutrals , molecules at low enough temps
metal definition for astronomy
anything other than H or He
why do H and He lines only occur in hot stars
helium and hydrogen have high ionisation/ excitation energies (to move e- to upper level) so can only occur in hot stars
why are metal lines formed over a large temperature range?
many species with a range of ionisation energies depending on atomic mass
ionised metals occur more in
hotter stars because electrons can be stripped from their atoms in hot temps
neutral metals occur more in
relatively cooler stars
why can metals only exist in the outer parts of cool stars (or in thermally insulated parts of hotter stars like sunspots)
they are broken up at fairly low temperatures
100s to a few 1000s of kelvin
in the Harvard classification scheme, each type (OBA etc) is split into
10 subgroups 0-9 with zero being the hottest
line strengths and equivalent widths vary with
spectral type
early attempts to classify stars used
the strength of their hydrogen lines
how can singly ionised helium be observed when mean energy of an electron «_space;ionisation energy of helium
maxwell boltzmann distribution
there exists some electrons with energy high enough to ionise helium
Hr-diagram, groups stars according to
spectral type and absolute magnitude
what did the first plot of a HR-diagram reveal?
a main band of stars
main sequence
problems with classifying stars using Hr-diagram
lack of fine resolution in the OBAFGKM scheme so stars line up in columns and data is discretised
HR classification is subjective based on ability to detect spectral features
some stars have odd spectra and don’t easily fit into any of the classes
because of difficulties with classifying stars using HR-diagram, often use
stellar colour instead as the basis of star classification
stellar colour
difference between stellar magnitudes
how is stellar colour measured
using standard filters in different wavelengths such as the Johnson UBV filters
eg B-V colour = mb-mv
difficulties with using stellar colour
requires well-calibrated filters for comparison between different instruments
can be tricky in a practical sense
colour is not independent of
distance
but spectra are
extinction of light due to interstellar medium along the observer’s line of sight is dependent on
wavelength
the diagrams essentially displaying the same info as HR are called
colour-magnitude diagrams
standard when plotting colour-magnitude diagrams is to use
UBV photometric system
Johnson system
U=m in U band (central wavelength =365nm, delta lambda=68nm)
B=m in B band (central wavelength = 440nm, delta lambda=98nm)
V=m in V band (central wavelength = 550nm, delta lambda=89nm)
colour index of a star
difference between the values of pairs of its Johnson magnitudes
eg U-B or B-V
can be extended to infrared with Cousins filters (UBVRI) (R=red, I=infrared)
transmission coefficient (used in Johnson-Cousins filter transmission profiles)
1= all light goes through
0= no light goes through
eg window approx 0.9
colour-colour diagram
eg U-B against B-V for objects of different temperatures
since filters produce brightness measurements in narrow frequency ranges,
a blackbody of a given temperature has a unique position on a CC diagram
colout colour diagram features
- main sequence lies below blackbody line
- there is a point of inflection between A5 and G0 stars
- the origin corresponds to a star of type A0V with (B-V)_0=0
the U-B axis on CC diagram increases
downwards for a given B-V colour so stars have U larger than B, they have a deficit of U band photons to B band photons
the large dip in photon flux at the blue end of the spectrum is due to
neutral hydrogen in the n=2 level being ionised
theis is the balmer series and the decrease in photon number shortward of 364.6nm is called the balmer jump. It is an ionisation edge
stars can be broadly classified according to their
luminosity
stefan boltzmann law
L prop to R^2T^4
for a given T, luminosity tells us about a star’s size
Morgan Keenan Kellman luminosity classification system
Class I: supergiants
Class II: intermediate giants
Class III: giants
Class IV: subgiants (for F-K stars), main sequence for B stars
Class V: main sequence
luminosity classification extends the classification of stars into
a two dimensional system of temperature and luminosity
classification of the sun
G2V
luminosity indicator
spectral feature (such as a line or a group of lines), the property of which is dependent upon the stellar luminosity or absolute magnitude, within one spectral class
luminosity indicator - O stars
width of the hydrogen lines increase with decreasing L
luminosity indicator - G stars
CaII, H, K widths increase with increasing L
luminosity indicator - K stars
CN line strength increases with increasing L
luminosity indicators are positive or negative depending on
whether the appearance of the indicator is enhanced or reduced as the luminosity increases
(enhanced =+, reduced=-)
photons travelling through the interstellar medium are
absorbed and scattered by it
this leads to extinction and reddening
extinction/reddening is a systematic shift of
a star’s position on the colour-colour or colour-magnitude diagrams
one of the disadvantages of using photometric instead of spectroscopic classifications
effects of extinction/reddening are dependent on
distance and can be corrected for
blue light and red light in the ISM
blue is scattered/absorbed by ISM gas/dust more than red light
bluer photons are removed and the star appears reddened. (interstellar reddening)
interstellar extinction causes the loss of photons
to quantify…
V - apparent measured magnitude in visual band
mv - the true apparent magnitude
Av - extinction coefficient
V=mv+Av
mv related to distance modulus
the amount of absorption depends on
the optical depth, tau, of dust in the ISM
Iout=Iine^-tau
so A=mout-min=-2.5log10e^-tau
optical depth is defined as
tau=N sigma d
N = number density
sigma = interaction cross-section
d=distance
for dust grains larger than lambda,
set sigma to be the grain’s geometrical cross-section
for dust grains smaller than lambda,
the interaction cross-section becomes dependent on wavelength and we observe reddening
blue light is scattered efficiently by
Rayleigh scattering where the scattering cross-section (probability a photon is scattered) varies as 1/lambda^4
due to the wavelength dependence of extinction, the B band flux
decreases more than V band flux
B-V and U-B both increase with increasing distance to star
measured B-V colour index must be
corrected by the colour excess to find true B-V colour of the star
Av is approx
3 E_B-V
bolometric correction
Mbol=MV+BC
BC can be determined from Pogson’s equation
how to find L and BC empirically
integrating over observations across a broad set of filters or using synthetic spectra
compared to solar type stars (spectral energy distribution peaks in visual band), bolometric corrections are
larger for early (hot, UV peak) and late (cool, IR peak) type stars
a light curve is produced by
monitoring the recorded intensity of a star over time (aka time series)
the light curve can be used to glean information about various aspects of the star such as
surface features on a rotating star such as starspots or disk hotspots.
the presence of a planet
stellar pulsations eg cepheid variables
eclipsing binary system: stars are so close to each other that
tidal forces cause mutual distortions away from sphericity
proportion of stars in binary systems
early-type stars >50%
solar-type stars approx 50%
late-type stars <50%
binary components not usually individually resolvable but using spectroscopy can look at
doppler shift of lines from spectroscopic binaries
light curves of eclipsing binaries
spectroscopy allows what measurements to be made
radial velocity
“radially towards the observer”
the inclination, i of a system with respect to our observation direction tells us
whether the system is edge on, i=90 i or face on i=0
edge on
i=90
bodies can totally occlude one another
face on
i=0
observer is above or below the system
(think plate analogy)
orbital period P of a star can be determined by
plotting the radial velocity curves of each star in the binary system
assume circular orbits, v=2pi r/P
a lower limit on the total on the total system mass M=m1+m2 can be obtained using
Kepler’s third law
assume momentum conservation
how to determine sin i and hence radii of stars
observe eclipsing stars
combine with radial velocity measurements
if stars are seen to totally eclipse then we may assume
i=90 so sini=1
we can infer stellar radii from
eclipsing binary light curves
realistic eclipsing binary light curves
must find a fit to i based on theoretical models of the light curve, including limb darkening
exoplanets have been discovered using three main methods
radial velocity measurements of the parent star
planetary transits
gravitational lensing
difficulties of radial velocity method
very high resolution spectroscopy and bright sources needed due to delta lambda approx 10^-14m
difficulties of transit method
very bright, stable sources are required so as to reduce the impact of photon counting noise and intrinsic variability in light curves
need regular long term observations of multiple transits
what have produced promising exoplanet candidates
big surveys eg kepler satellite which monitor hundreds of thousand of stars and focus on transits
separate radial velocity measurements can be done to determine mass, radius of orbit and planet density
when the transit light curve is used to determine the radius of the planet…
density and surface gravity can also be found
assuming sphericity and constant density
from the planet’s orbital radius, can determine
if a planet is in the habitable zone (region where liquid water may exist)
after expending some fraction of their core fuel, stars
evolve off the main sequence
as stars move towards the giant phase
they undergo pulsations
cepheid variables
used as distance indicators
pulsates on a regular basis such that its radius changes
as a consequence, there are also changes in luminosity L=4piR^2sigmaT^4
cepheids are used as distance indicators because
they have a tight correlation between oscillation period and luminosity
cepheid variable: what forces are at play?
driving force and the restoring force
driving: radiative force drives outer layers of star outwards
restoring: gravity pulls them back
we can derive the period-luminosity relationship by analogy with the case of
the simple pendulum
start from P=2pi sqrt(l/g)
period-luminosity relationship derivation from simple pendulum
replace l with R and g=GM/R^2
assume spherical with mean density
then period mean density relationship
period-luminosity relationship can be sued to determine
the absolute magnitude and hence distance to a Cepheid and its host galaxy (what makes them standard candles)
the oscillation (oulsating cycle) only happens under the circumstances
particular to evolved stars
to maintain the cycle, requires that the radiative driving term
decreases as the star expands and increases as it contracts
the effectiveness of the radiative driving term depends on
how opaque the as is, described by the opacity kapa
high opacity
large push from radiative driving
low opacity
photons stream right through; no push
cepheid cycle diagram
atmosphere contracts
ionisation increases (everything closer=more collisions)
opacity increases
radiation trapped
large outward radiation force
atmosphere expands
ionisation decreases
opacity decreases
radiation streams out
gravity dominates
the baade-wesselink is used to
calculate the radii of pulsating stars
we require to know the variation of magnitude, stellar colour and radial velocity with time
Baade-wesselink: pick data times t1 and t2 where
the colour indices are approximately equal
this means stellar temperatures are also approximately equal
Baade-Wesselink: luminosities are different because
radii are different
two equations from Baade-wesselink
- from stefan boltzmann and pogson
- from spectroscopy we also have radial velocity as a function of time
Baade wesselink: factor of 3/2 arises because
we measure vr averaged over the stellar disk, it is the average of vrcostheta from theta=-pi/2 to pi/2
spectral properties of hot stars are dominated by their
huge radiation output
they are massive stars with circumstellar winds
circumstellar winds are driven away by
radiation pressure
the outflows of gas result in a characteristic spectral line shape - a P Cygni profile
thomson cross-section, sigmaT
consider electron at distance d from a star of luminosity L
electron has a probability of interacting with a photon which is described by its interaction cross-section
we can think of photon flux as a momentum flux since
each photon carries momentum
when a photon strikes an electron, and is absorbed or scattered momentum conserved so electron momentum increases
thomson cross section derivation
start with flux and energy of a photon
number of photons per second per square meter = f/E
p=hv/c
radiation force on electron is equivalent to momentum transferred per unit time
electron will move outwards from the star if
Frad > Fg
Fg acts on everything but since mp»me
its relative effect comapred to Frad is bigger for protons compared to electrons
Eddington luminosity
luminosity above which a star will blow away its outer layers in a wind
mass loss rates are deduced from
observations and by applying the mass continuity equation so as to calculate the density and velocity of the stellar wind
mass loss rates are conserved
mass leaves the shell at the same rate it enters
all hot stars have winds driven by radiation pressure directly inferred by
the presence of a P Cygni profile in the star’s spectrum
two main characteristics of P-Cygni profile
blue shifted absorption dip
red wing emission
blue shifted absorption dip cause by
absorption of the star’s radiation by the stellar wind moving towards us, between us and the star.
red wing emission caused by
light scattered towards us by particles in the wind and light that is directly emitted by the stellar wind
wind speed increases with distance from the star up to
some maximum speed known as the terminal velocity
in terms of a P Cygni profile, terminal velocity is defined as
the point where the absorption dip crosses the velocity axis
velocity of stellar wind increases with
increasing radial distance according to stellar wind velocity law
it is possible to measure from the P Cygni profile
velocity at the surface of the star is close to the local sound speed and depend only on stellar surface temperature
main early type (OBA) stars are rapid rotators
evidence for this is seen in
their broad, (semi-) elliptical line profiles
if you observe a large value for delta lambda, you may conclude that
the star’s rotation speed is also large
the combination of many rotating components towards, at rest and away from the observer…
produces a broad profile with half-width delta lambda
total width of profile is combination of blue-shift, rest and red-shift
all points on a chord AB, distance l from the rotation axis are travelling at the same speed and they produce
Doppler-shifted absorption or emission
for each individual line component, the line Doppler shift depends on
the chord’s l position only
- l=R - delta lambda (l) = max
- l=0 - delta lambda (l) = 0
the strength of the spectral line depends on
how much material is contributing to it
I(delta lambda) / I (lambda 0) = AB/2R
If the star rotates with its axis at some inclination angle i then VE can be replaced by
VEsini
the maximum rotation period of a star can be calculated since
VE>VEsin i
Be stars
B stars with hydrogen balmer lines in emission
eg balmer H alpha (level 3 to 2 transition)
the emission of Hbeta in a Be star is thought to originate from
a circumstellar disk of material
Be stars are surrounded by
disks or have extended envelopes
depending on the viewing angle of the star, the observer sees
different line shapes
B stars exhibit strong radiation pressure. Since they are also rotating, the material on their surfaces is
subject to a strong centrifugal force which helps to drive material outwards from equatorial regions of the star, forming a disk
what shows Be stars to have particularly pronounced rotation
broad elliptical profiles
a particle in a stellar atmosphere rotating with the star is subject to
radiation, centrifugal and gravitational forces
how to determine conditions where equilibrium breaks down
equating forces
eg: equation centrifugal and gravitational gives maximum rotation velocity
centrifugal force has the biggest effect at
the equator where theta=pi/2
why can material escape more easily from equator
lower effective surface gravity at the equator due to rotation
centrifugal break-up limit
Fcen + Fg > or = 0
maximum rotation velocity possible before the star breaks up
set RHS=0 and theta=90 (equator)
thomson scattering occurs when
photons are scattered by electrons which remain bound during the collision
predominant scattering process in B and Be stars
thomson scattering
thomson scattering in the corona
photospheric radiation is Thomson scattered by electrons spiralling in the Sun’s coronal magnetic field and it is this that we observe during a solar eclipse.
polarisation describes how
electric and magnetic components of the electric field behave relative to its propagation direction
if we look at only the electric field, the act of Thomson scattering is similar to
that of the polarising filter
after filter, electric field oscillates in a single direction and is linearly polarised
circular polarisation
special case of elliptical polarisation arising when x and y electric field vectors are of equal magnitude and out of phase by pi/2
in stars, a polarisation signal tells us
about the circumstellae material because therein is what gives rise to polarisation
polarisation signal is dependent on
the shape of circumstellar material
spherically symmetric wind gives
zero net polarisation
irregular envelope can give
polarisation signal that is dependent on the shape
disk tends to give
strong linear polarisation
studies of spectral line profiles and polarisation in Be stars show
linear polarisation, supporting the theory that these stars have circumstellar disks
stellar classes F to M show evidence for
magnetic activity
sunspots
surface magnetic field
activity cycle
rotation
flares
coronal magnetism
surface oscillations
interior magnetic field generation
helioseismology
zeeman splitting
magnetic field strengths
calcium k line reversal
average field strengths
activity cycle
doppler imaging
surface starspots
zeeman splitting occurs because
in a magnetic field, the energy levels of the electrons in an atom depend on their orbital angular momentum
the level of splitting depends on
the strength of the magnetic field
+/- in zeeman splitting equation
reflects that the line is split with one component at shorter and one at longer wavelengths (either side of original position)
ballpark figure for zeeman splitting magnetic field
1T
considered very strong
stellar magnetic field is produced in
the stellar interior
from zeeman splitting know starspots are locations of strong magnetic fields
sunspots and starspots are formed when
loops of magnetic field beak through the stellar surface
direction of field lines
direction of the magnetic field
number of field lines
prop. to its strength
starspots appear dark because
they are cooler than the surrounding material
a strong magnetic field will inhibit heat transport
cooler or hotter patches on the surface of a rotating magnetised star will
produce photometric variations which are quasiperiodic in nature
quasiperiodic
periodic just not perfect sin
fractional area
tells you how much of the surface is sunspots
f=1, whole star is sunspots
f=0, no spots
total radiation brightness from visible disk of star
B=photosphere + spot
fractional area of sun
around 0.5%
cool sunspots will reduce
the total brightness of visible disk hence reducing the flux at earth and increasing the observed magnitude
(B1=spots, B2=no spots)
the presence of a starspot also changes the
rotationally broadened absorption line shape
a spot at longitude theta will show up as what in the absorption line
bump or dip
feature moves through the line as the spot rotates across the disk
a star’s observed flux changes as
the star rotates in the presence of starspots
doppler imaging can be used to
work out the distribution of light and dark patches on the star
if projected rotational velocity too slow
not enough spatial resolution for good imaging
if projected rotational velocities are too fast
lines become too shallow due to spread from doppler broadening to make out effects due to spots
doppler shift at i=0
no rotational doppler shift
doppler shift for i=90
mirror symmetry about the equator so not possible so not possible to determine if spot is in northern or southern hemisphere
most familiar signature of solar magnetic fields
sunspots (or starspots) visible in broadband white light
evidence for solar magnetic fields observed in regions outwith sunspots and starspots
K line of Ca II
because of its close association with regions of strong magnetic field in the sun the Ca II K line is used to
infer magnetic fields in other (solar-like) stars
can also be used as a luminosity indicator
sunspots where the
strong magnetic field lines are
calcium II K line
in emission with wings in absorption and then reabsorption in the cntre
K1
absorption of BB radiation by Ca II ions in the photosphere gives an absorption line (K1)
K2
hot chromosphere above the photosphere gives line-centre emission (K2)
K3
re-absorption and scattering in the chromosphere gives a central re-reversal (K3)
on the sun the ratio of core to wing emission in the Ca II K line varies with
magnetic field strength
the correlation is a straight line in a log-log plot
for the sun the total number of sunspots varies over a cycle of
approx 11 years
during cycle the K2 component of the Ca II K line varies also
K2 emission requires a
chromosphere (hotter,tenuous gas overlying a region of cooler, denser gas)
wilson-bappu effect
describes the relationship between the Ca II K2 line width W and the magnitude of stars
K2 line width W correlates linearly with absolute visual magnitude over 15 magnitudes
flares
most dramatic signs that a star has a magnetic field
sudden and transient releases of energy in the solar atmosphere
flares are strong, sudden brightness variations at shorter wavelengths
what approx % of sun’s surface do flares occupy?
0.002%
flare rating system
each letter (ABCMX) represents a factor of 10 increase in power
a loop arises when
hot filament of charged material rises from the Sun’s surface before magnetically reconnecting
energy transport in flares
loop arises
magnetic field lines extend beyond material
cool plasma moves in from sides
hot plasma flows up and down
storage of energy released as the flare
Dwarf M stars with Balmer lines in emission release up to
100 times the energy of a typical solar flare
around 2% of stellar surface involved in flare
the interior temp and density structure of a star depends on its
mass which in turn determines how energy is transported out into space
energy generated in the core by
fusion
this is converted into EK of particles plus photons and neutrinos
random walk
density of solar interior very high so photons constantly collide with particles and move in a random walk
in solar type stars gas is transparent enough for photons to
carry all energy outwards until opacity of gas increases enough that convection will take over
tachocline
boundary between radiative and convective zones
where sun’s spin rate changes dramatically
rapidly flowing ionised gas produces strong electrical currents and gives rise to magnetic field
late type stars are thought to be
fully convective
imaging of M giant show surface intensity is
not uniform
has one or more giant hotspots
interpretation is that hotsopts are large convection cells
when the stellar gas becomes opaque to radiation…
energy is trapped
at layer where energy is trapped, gas would become very hot if other process did not take over to move energy
temperature gradient
rate of change of temperature with distance
convection stars when
temperature gradient becomes high enough
convection is
the transfer of heat by bulk flows in a hot material
for convection to happen in a fluid or gas
hot material must be able to rise and cool material sink
convection occurs if an element of gas in a stellar interior continues to rise upwards when displaced. This happens when
the gas bubble is buoyant
equation governing thermodynamics for convection tells us that
if stellar temperature gradient increases to the critical value defined by the adiabatic temperature gradient, convection starts
the adiabatic temperature gradient can be calculated once the
stellar pressure gradient is known
solar granulation
surface of stars with a convective layer like the sun have a surface pattern that looks like a pan of boiling water
on the sun individual granules are how big
700km-2000km across
lifetime of solar granules
about one hour
centre of a granule
slightly brighter and therefore hotter than the average
continuous appearance and disapperance of cells is a
random process and leads to a small-scale fluctuation in the solar brightness
calculating fractional brightness increase due to any bright cell appearing
delta B / B
tempcell^4-tempstar^4 / tempstar^4
red dwarfs
late type stars that are cool enough for molecules and dust to survive around them
show evident for CO, CN and CH4 and more complex molecules.
simple molecules are thought to coagulate
into dust which can be driven by radiation pressure into the interstellar medium
debris around a red dwarf
not enough radiation force from the star to clear the dust
brown dwarfs
in main sequence stars, hydrogen burning is taking place
at lower temps there are cooler, ‘failed stars’ known as brown dwarfs
brown dwarfs have never
gone through a main sequence stage
may have briefly undergone lithium or deuterium fusion
brown dwarfs are identified based on
evidence for molecules in their atmosphere
(bottom right of HR diagram)
decreasing temp implies
increasing molecular complexity
wien’s displacement law tells us that spectral bands are in IR region
imaging at different IR wavelengths would allow us to
probe different layers in brown dwarfs atmosphere because weather should be observable due to cloud structure causing doppler broadening of spectral lines
stellar formation
stars condense out of gas clouds which contract under their own gravity (if massive enough)
as gas cloud contracts it loses gravitational potential energy which heats up gas and ejected in bipolar flows away from the protostar
when is a star not a star?
objects with mass not much more than jupiter can reach high enough T for deuterium burning - brown dwarf
more mass allows hydrogen burning for a red dwarf or main sequence star to form.
