Time Domain Astronomy
Astronomy is the study of the characteristics of Astronomical
Objects
●Location on the sky
●Morphology
●Brightness
●Spectrum
●Physical characteristics
●Mass
●Gravity
●Chemical composition
●Magnetic field, etc.
Primarily a study of static characteristics
Time Domain Astronomy
Some (all?) things do change:
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Location on the Sky
–
Seasonal motions of objects
–
Orbital motion of solar system objects
–
Proper motion of stars
Time Domain Astronomy
Some (all?) things do change:
●
Location on the Sky
–
Seasonal motions of objects
–
Orbital motion of solar system objects
–
Proper motion of stars
–
Parallax of stars
●
The first leg in the cosmic distance scale
Parallax
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The distance measure Parsec is based on parallax
measurements and is defined as the distance at which an
object exhibits 1” of parallax as seen across the earth's orbit
Time Domain Astronomy
Some (all?) things do change:
●
Location on the Sky
–
Seasonal motions of objects
–
Orbital motion of solar system objects
–
Proper motion of stars
–
Parallax of stars
The first leg in the cosmic distance scale
Physical characteristics (evolution)
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Morphology
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Brightness (Luminosity/Magnitude)
Variable Stars
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Galactic and solar system objects vary as well
–
AGN's
–
Asteroids (tumbling)
–
Rotation/Orbital motion exposing different surfaces
–
Seasonal changes to surfaces
–
Occultations
Occultations
Occultations
Occultations
Occultations
Occultations
Occultations
Variable Stars
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Variable stars – Three broad classifications:
–
Extrinsic variables
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Occulations
Variation from rotation, non-spherical stars, star spots
Magnetic field variables
Eclipsing binaries, star-star, star-planet
Variable Stars
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Variable stars – Three broad classifications:
–
Cataclysmic/Eruptive variables
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Proto-stars
Flare Stars
RS Can Van
Wolf-Rayet Stars
Novae/Dwarf Novae
SN (3rd leg in the cosmic distance scale)
Pulsating Variables
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Irregular/Long period variables (Mira)
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Solar-type oscillations
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Instability Strip Variables
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–
Delta Scuti types (inc. roAP, SX Phoenicis)
–
RR Lyrae (2nd leg in the cosmic distance scale with:)
–
Cepheids
RR Lyr and Cepheid variables are post giant stars moving bac
across the H-R diagram and passing through the instability st
Standard Candles
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Cepheids and RR Lyr establish distances out to galactic
distances
RR Lyr are more numerous, but fainter and are used for
distances out to globular clusters and nearby galaxies
Standard Candles
RR Lyr Stars in the globular cluster M13
Standard Candles
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RR Lyr evolving through the horizontal branch and instability strip
For RR Lry stars luminosity is nearly constant as stars get smaller
and hotter at about Mv=0.7
Standard Candles
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Cepheids and RR Lyr overlap establishes distances out to
nearby galaxies
M33 Cepheids and Hubble
Standard Candles
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Distinct period-luminosity relation for Cepheids can be used to
establish distances well beyond the reach of RR Lyr stars onc
the absolute magnitude scale is fixed
Standard Candles
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The Distance Modulus
μ=m−M
M =−2.5log ( L(10 PC ))
m =−2.5log ( L( d ))
–
Luminosity varies as inverse square of distance
L ( d )=
L (10
(
–
–
PC
2
)
d
)
10
Combining gives μ=m−M =5log (d )−5
Finally
d PC =10
μ
+1
5
Period Finding
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Two fundamental techniques for period analys
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Fourier Transforms
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Reconstruction of time series data with a sum of
sinusoids of different frequency and amplitude
The Fourier Transform
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Used primarily for well sampled data, i.e. data with many
samples during one cycle of the periodicity searched for
Nyquist frequency is the largest frequency that can be
unambiguously determined by a data set
–
1
1
F samp =
2
2∗dT
F(Nyquist) =
where dT is the smallest or
fundamental sampling interval in the data
The Fourier Transform
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Single sinusoids transform into delta functions in the FT if
the data are infinite and evenly sampled
Any deviation from that condition will throw power away
from the primary frequency
Evenly sampled data of finite length (real world) are a
multiplication of a sinusoid and a box function in time, so
are a convolution of a delta function and a sinc function in
frequency space
There is a minimum number of frequencies needed to
reconstruct the data. This is the “Number of Independent
Frequencies”, and is set by the number of data points.
More data points will increase the true frequency resolution
of the data set in the FT
The Fourier Transform
The Fourier Transform
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DC components and data trends should be removed to
eliminate low frequency power which will spill over into
frequencies of interest
Number of data samples controls how many independent
frequencies are available to fit the data.
–
Overall shape of transform at the period of the signal is the
same, but longer data set with more points allows for more
precision in the measured frequency.
The Fourier Transform
The Fourier Transform
●
●
●
DC components and data trends should be removed to
eliminate low frequency power which will spill over into
frequencies of interest
Number of data samples controls how many independent
frequencies are available to fit the data.
Noise that is truly random cannot by definition have a preferre
frequency. This means that Gaussian noise has to be
distributed overALL independent frequencies in the FT
The Fourier Transform
The Fourier Transform
●
●
●
●
DC components and data trends should be removed to
eliminate low frequency power which will spill over into
frequencies of interest
Number of data samples controls how many independent
frequencies are available to fit the data.
Noise that is truly random cannot by definition have a preferred
frequency. This means that Gaussian noise has to be
distributed overALL independent frequencies in the FT
Gaps in the data produce ambiguities in the FT's selection of
where power goes. Periodic gaps will produce
aliasesof the
real frequency separated by the period of the gaps. The most
common is 1/day aliases due to diurnal gaps from the sun
spoiling our fun
The Fourier Transform
The Fourier Transform
●
●
●
●
●
●
DC components and data trends should be removed to eliminate low
frequency power which will spill over into frequencies of interest
Number of data samples controls how many independent
frequencies are available to fit the data.
Noise that is truly random cannot by definition have a preferred
frequency. This means that Gaussian noise has to be distributed
over ALL independent frequencies in the FT
Gaps in the data produce ambiguities in the FT's selection of where
power goes. Periodic gaps will produce aliases of the real
frequency separate by the period of the gaps. The most common is
1/day aliases due to diurnal gaps from the sun spoiling our fun
The Beam or Window of the data is a FT of the sampling pattern with
a single clean sinusoid.
The Window for the data set can be used to isolate potential multiple
periodicities in the data
The Fourier Transform
The Fourier Transform
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The Beam or Window transform of the data is a FT of the
sampling points with a single clean sinusoid.
The Window for the data set can be used to isolate potential
multiple periodicities in the data
Frequency points in a FT are actually vectors in the complex plane
Lengths of the vectors are the power or amplitude of the periodic
signal
Rotation of the vector in the plane can be retrieved from the
real/complex components and is the phase of the sinusoid running
through the data
Pick the peak amplitude and subtract a sinusoid from the data
with the same amplitude and phase.
This is Pre-Whitening and any remaining power that matches the
window transform is a second period
Phase Dispersion Minimization
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This technique developed to find periods in light curves that are
poorly sampled (Nyquist), with large gaps, or having non-sinusoidal
shapes
These problems can make Fourier analysis difficult either through
Nyquist sampling aliasing, aliasing or power from non-sinusoidal
shapes
Data are folded at different periods and put into a set of bins covering
the full 0.0->1.0 of the phase of the period
Dispersions between the data points are ratioed to the dispersion of
the data set as a whole
Minimums in the dispersion are true periods or sub-harmonic aliases,
usually the deepest minimum is the true period
Phase Dispersion Minimization
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PDM works in two steps
–
Coarse frequency steps to find likely period minimum
–
Fine frequency steps around likely period to establish
precise frequency, and fitting of folded curve
Typical RR Lyr data set
Phase Dispersion Minimization
The folded light curve and Theta plot
Phase Dispersion Minimization
If light curve has multiple maxima, likely a sub-harmonic was
chosen, and PDM will have to be redirected to the correct
period. Examine each selection for a single peaked light curv
Multi-periodic RR Lyr Stars
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RR Lyr can exhibit multiple modes of pulsation
Two period will beat against each other producing a
fundamental period with amplitude modulate by the beat
f B e a=
t f 1− f 2
frequency