Astro497_2 Henry Yul Doppler

7/27/2019 Astro497_2 Henry Yul Doppler

1/12

Doppler spectroscopy and

astrometry

Theory and practice of planetary orbit

measurements


2/12

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


7/12

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


10/12

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

Astro497_2 Henry Yul Doppler

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