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Astro497_2 Henry Yul Doppler
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Transcript of Astro497_2 Henry Yul Doppler
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Doppler spectroscopy and
astrometry
Theory and practice of planetary orbit
measurements
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Geometry of a binary orbit
P
x
z
y
Z
i
i
To Earth
Line of
nodes
i - orbital inclination
- longitude of
periastron- true anomaly
P- periastron
r1=
a11" e
2( )1+ ecos#
x = r1cos"
y = r1sin"
z = r1sin"sini
"=#+ $
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Orbital elements
A binary orbit is defined by 7 elements:
Size: a = a1+ a2 semi-major axes of the orbits
Shape: e eccentricity
Orientation in space: i, , (longitude of periastron)
Location in time: T time of periastron passage, P
orbital period
In Doppler spectroscopy, five orbital elements (a1, e,
, P, T) can be determined from radial velocitymeasurements of one binary companion
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Determination of radial velocities
Vr= z = sini[rsin("+#)+ r"cos("+#)]
r=
esin"r"1+ ecos"
r2"= 2#a2(1$ e2)1/ 2
P
Vr=
2"asini
P 1# e2
cos($+%) + ecos(%)[ ]
Radial velocity, Vr, is a time derivative of a component ofthe radius vector along the z-axis:
Time derivatives ofri can be computed from the equation
of elliptical motion and from the 2nd Kepler Law:
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Examples of radial velocity curves
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Models of orbits from Vrmeasurements
Observations are given in the form of a time series, Vr(i),at epochs t(i), i = 1,,n
A transition from t(i) to (i) is accomplished in two steps:
E" esinE =2#
P (t"T)
tan$
2
%
&'
(
)*=
1+ e
1" e
%
&'
(
)*
1/ 2
tanE
2
%
&'
(
)*
Vr = K cos($++)+ ecos+( )
Equation for
mean anomaly, E
K=2"a
1sini
P 1# e2
From the fit (least squares, etc.), one determines parameters K, e, , T, P
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Planetary mass determination
From K= (Vmax - Vmin)/2 we get:
a1sini =
1" e2
2#KP
A planetary mass (times sin i) is found by assuming that
the mass of the star is known:
f(m2) =m2
3sin
3i
M2
=
(1" e2)3 / 2K
3P
2#G
For a fixed amplitude Vr
m2sini~ a1/2
detectable
not detectable
log a
logm2sini
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Applicability and limitations of Doppler
spectroscopy
The methods allows determination of 5 out of 7 parameters of the orbitprojected onto the sky plane. Without an independent measurement ofi, one gets only a lower limit to the mass of the planet
Ability to measure very small changes of Vrare necessary (e.g. Jupiter- 12,5 m s-1, Earth - 0,1 m s-1, a spectrometer with the resolution of
R=105
allows to measure Vron the order of 10-5
c ~ a few km s-1
) Photon noise (uncertainty of flux estimate ~N-1/2/pixel) provides an
absolute limit of the precision of Vrmeasurement
Measurements of large numbers of lines improves the signal-to-noiseratio, (S/N)~ (# of lines)1/2, S/N depends on the spectral type of star
For a G star with V=8, S/N ~ 200 can be achieved with a 3-mtelescope. This gives a theoretical Vrprecision ~ 1-3 m s
-1
Another practical precision limitation results from stellar activity. Thisproblem can be controlled to some extent by modeling. Currentlyattainable precision is ~3 m s-1 (Saturn mass for G-stars 10-20 massesof Neptune for K,M dwarfs)
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Calibration, analysis and examples of Vrcurves
Stellarlight
I2 cell
spectrometer
CCD
Modern observing hardware andtechniques of spectral analysis
allow Vrmeasurements at a ~10-3
pixel precision
Analysis is done by cross-correlating
the spectra with high-S/N templates,
the use of many spectral lines, and by
accurate calibration with of I2 cells
installed in the optical path of
the telescope
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Astrometry
P
x
z
y
Z
i
i
r1=
a11" e
2( )1+ ecos#
x = r1cos"
y = r1sin"
z = r1sin"sini
"=#+ $
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Astrometry: basic characteristics - I
Astrometry measures stellarpositions and uses them to
determine a binary orbit projected
onto the plane of the sky
Astrometry measures all 7
parameters of the orbit
In analysis, one has to take theproper motion and the stellar
parallax into account
The measured amplitude of the
orbital motion is simply a1=
(m2/m1)a. Assuming m2
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Astrometry: basic characteristics - II
"#= 0.5q
10$3
%
&'
(
)*
a
5AU
%
&'
(
)*
d
10pc
%
&'
(
)*
$1A comparison of some
astrometry situations
Takingq=m2/m1,we can calibratethe expression for:
A unit foris one millisecondof arc - very small effect
Amplitude of the effect depends
directly on d
Dependence of m2 on a is opposite
to that in Doppler spectroscopy
detectable
not detectable
Log a
Log m2