Description of the Doppler Method

Currently, there are two major Doppler methods: one using high resolution cross-dispersed echelle spectrographs (the echelle method) and the other using dispersed fixed-delay interferometers (the DFDI method). Both methods have been successfully used for detecting new planets (e.g., Butler et al. 2006 for a summary of exoplanets detected by the Doppler techniques). Here we briefly describe both methods.

2.2.1 The High Resolution Cross-Dispersed Echelle Method

The RV method using high resolution optical spectrographs was proposed for detecting extrasolar planets in the 1950's by monitoring spectral line shifts of the target star caused by the gravitational pull of the planets (Struve 1952). However, it was difficult to obtain better than several tenths of a kilometer per second precision before the 1970's due to the use of inadequate instrument calibration methods, in which the reference beam does not follow the same optical path as the stellar beam (Griffin 1967). In order to detect Jupiter like planets, a Doppler precision of ~ 10 m/s is required (e.g., the velocity semi-amplitude of the Sun caused by the gravitational pull of Jupiter is about 12.3 m/s over 11.86 years).

In 1973, Griffin and Griffin proposed to use telluric absorption lines as a reference for RV measurements to eliminate the differential motion between the reference and stellar beams in order to achieve high RV precision (Griffin & Griffin 1973). This method was further developed by Campbell & Walker in the late 1970's (Campbell & Walker 1979). Instead of using telluric absorption lines, which vary and also shift slightly due to line saturation and atmospheric winds, they used a toxic hydrogen fluoide (HF) gas cell which produces the R branch of the 3-0 vibration rotation band in the wavelength region of 8650-8800 A (Campbell & Walker 1979). A Doppler precision of ~15 m/s was achieved. Walker et al. monitored a total of 21 bright solar-type stars using this method and a Coude spectrograph on the Canada-France-Hawaii 3.6-m telescope (CFHT) over 12 years. Although they did not detect any exoplanets, their results indicate that less than 5% of solar-type stars have planets larger than 2 Jupiter masses within 5 AU (Walker et al. 1995). This result is consistent with the recent conclusion based on the California-Carnegie planet survey (Marcy et al. 2005).

In the late 1980s, molecular iodine was chosen to use as a reference instead of HF. Unlike HF, molecular iodine is non-toxic and has thousands of absorption lines in the wavelength region of 5000-6200 A, which can be used for tracking the instrument velocity drifts and also instrument response changes. Another calibration method using a stabilized Fabry-Perot etalon interferometer was also developed at the same time and has reached a precision of ~ 8 m/s over ~ 5 years (McMillan et al. 1993). The iodine absorption calibration method was used to achieve ^3 m/s Doppler precision with the Lick Hamilton echelle spectrograph in the 1990s (Butler et al. 1996). Several other groups have also achieved similar Doppler precision using the iodine calibration method (Cochran & Hatzes 1993; Brown et al. 1994). The iodine calibration method has become popular for high precision RV measurements since then.

In the early 1990s, the ThAr separate beam calibration method reached a milestone, delivering ~ 10 m/s Doppler precision and long term stability using fibers for feeding the star and calibration light to the spectrograph (Baranne et al. 1996). The first extrasolar planet around a solar-type star, 51 Peg, was detected by this method (Mayor & Queloz 1995). Fiber feeding has played a key role in allowing the instrument to be installed in an isothermal environment to increase the instrument thermal and mechanical stability.

2 Doppler Exoplanet Surveys: From Single Object to Multiple Objects 23 Intensity

Fig. 2.1. Echelle working principle for Doppler RV measurements. SXi is the intrinsic line width of the absorption line, Di is the absorption line depth.

In the cross-dispersed echelle method, the Doppler precision is fundamentally limited by the total number of photons collected by the spectrograph. Figure 2.1 shows the principle for RV measurements using a high resolution echelle spectrograph. The photon-limited Doppler precision of an echelle instrument can be described as

where N is the total number of pixels, ej is the uncertainty in the intensity at pixel i, and dl/dV is the local slope of the absorption line (Butler et al. 1996).

For a fully resolved absorption line, the derived intrinsic Doppler measurement error due to the photon noise is:

cSXj

where the total photon number collected by each line is F = nASAiAtS, n is the total detection efficiency from the telescope to the detector, A is the telescope photon collecting area, At is the total exposure time and S is the stellar flux in photons cm"2 s"1 A"1. For an echelle with a resolving power of R = A/SAo, where SAo is the spectral purity of the spectrograph, the total measured photon-noise limited Doppler error for an absorption line can be approximately described as

SAi when SXo > SXi. Therefore, the total Doppler error over a wave band of AX can be approximated as aRV <x SXiS-0-5AX-0-5R-1-5D-1, (2.5)

(also see Hatzes & Cochran, 1992). This formula shows that the echelle Doppler precision strongly depends on spectral resolution (-3/2 power of the spectral resolution) and is also related to the wavelength coverage and stellar flux (-1/2 power of the wavelength coverage and stellar flux). This is the main reason that most of the planet hunting echelle spectrographs have a spectral resolution R >60,000 at optical wavelengths since the typical width of stellar absorption lines for a solartype star is a few km/s. The Doppler precision also depends on stellar properties: a star with deep and narrow lines (such as late type stars) tends to produce a higher Doppler precision than a star with shallow and broad lines (such as early type stars) using the same spectrograph with the same exposure time. This is one of the main reasons that current optical Doppler planet surveys are mainly focused on late type stars.

Currently, two major calibration methods are used in the cross-dispersed echelle instruments for measuring the instrument velocity drifts: the iodine absorption cell and the ThAr emission lamp. These two methods have their own advantages and disadvantages. Some advantages of iodine cell calibration are that (1) thousands of iodine absorption lines are superimposed on top of the stellar absorption lines; (2) both the starlight and iodine absorption share a common optical path, so the iodine absorption lines simultaneously track changes in the instrument point spread function due to the same physical effects causing instrumental drifts affecting the stellar absorption lines. Therefore, iodine cell calibration enables reaching photon-noise limited Doppler precision. However, the major limitation for the iodine method is that iodine has absorption line bands clustered in the visible (5000-6200 A) and also the absorption cell absorbs about 30% of the incoming photons. These limit the application of the iodine method for mainly observing relatively bright solar-type stars which have peak fluxes around the visible. The method becomes less efficient for late type stars such as M dwarfs which have peak fluxes at wavelengths much longer than the visible.

The main advantage of using the ThAr calibration method is that the ThAr lamp has hundreds of strong emission lines over the entire optical and near-IR wavelength range (Palmer & Engleman 1983; Hinkle et al. 2001) so it can be used for RV measurements over a wavelength band much broader than the iodine calibration technique. For instance, the High-Accuracy Radial velocity Planetary Searcher (HARPS) uses the entire 380-690 nm for RV measurements. In addition, since the calibration beam is separated from the stellar beam, no stellar photons are absorbed by the calibration optics, increasing the RV measurement throughput by ^30%. No overhead time is required to take the star and iodine templates during observation, increasing the observation efficiency. However, the main drawback for the ThAr method is that the entire instrument needs to be installed in an isothermal and mechanically stable environment, possibly even in a vacuum chamber such as HARPS, in order to minimize the differential movement between the reference beam and stellar beam. Also, fibers must be used in order to minimize the differential motion between the incoming stellar and ThAr beams. In HARPS, a fiber mode scrambler is applied to further reduce the fiber illumination variation caused by the seeing and the fiber guiding changes in order to reach sub m/s RV precision (Pepe et al. 2000; Rupprecht et al. 2004). Therefore, the instrument design becomes more complicated than that using the iodine calibration method.

2.2.2 The Dispersed Fixed-Delay Interferometer Method

The RV method using a Michelson type interferometer was proposed in the late 1970s (Gorskii & Lebedev 1977, and Beckers & Brown 1978). The Doppler shifts of the incoming spectral lines are measured through monitoring the interference fringe phase shifts. For a Michelson interferometer with an optical path difference d between the two interferometer arms, the fringe order m is determined by mX = d. (2.6)

For a small wavelength shift, SX, the Doppler shift, AV, can be derived as

where $ = 2nm is the phase of the interferences (Ge et al. 2002).

This kind of interferometer with a narrow band pass has been successfully used for very high precision Doppler measurements of the Sun (e.g., sub m/s precision for the Global Oscillation Network Group (GONG) measurements, Harvey 2002 private communications; Harvey et al. 1996). This narrow-band Michelson type interferometer with a fixed delay is suitable for observing bright sources such as the Sun, but not for faint targets such as other stars.

In 1997, David Erskine of Lawrence Livermore National Lab proposed to use a combination of a Michelson type interferometer with a moderate resolution spectro-graph for stellar RV measurements. The addition of the spectrograph separates the interference fringes at different wavelengths to increase the fringe visibility (or contrast) for each absorption line and the wavelength band in order to obtain high precision RV measurements for faint sources such as stars. Figure 2.2 shows a schematic layout of this kind of instrument concept (Erskine & Ge 2000; Ge 2002; Ge et al. 2002). The fringe visibility is defined as

±max \ ±mm where I is the fringe intensity. High-precision RV measurements can be achieved by summing independent RV measurements over many different spectral lines within the instrument wavelength coverage. This approach is called the dispersed fixed delay inferferometer (DFDI) method. The initial lab experiments and telescope observing with a prototype at the Lick 1 m telescope demonstrated its feasibility for stellar RV measurements (Erskine & Ge 2000; Ge et al. 2002).

Detector

Fig. 2.2. Principle of a dispersed fixed delay ineterferometer, a combination of a Michel-son interferometer with a moderate dispersion spectrograph. The spectrograph separates fringes from different wavelengths to allow high precision RV measurements using a broadband spectrum.

Detector

Fig. 2.2. Principle of a dispersed fixed delay ineterferometer, a combination of a Michel-son interferometer with a moderate dispersion spectrograph. The spectrograph separates fringes from different wavelengths to allow high precision RV measurements using a broadband spectrum.

In this approach, the photon-limited Doppler precision is described as vrv = , 1 —, (2.9)

where y* is the fringe visibility, and Fi is the photon flux in each of the N wavelength channels (Figure 2.3, Ge 2002; van Eyken et al. 2004). For a Gaussian-shaped absorption line, the intrinsic Doppler precision can be derived as f i K-S*** , (2.11)

which is the same as that for the echelle spectrograph (Ge 2002). This is not surprising since the intrinsic Doppler precision is totally determined by the spectral line intrinsic properties, irrelevant of measurement methods. However, when a spectrograph with a spectral resolution, SXo > SX*, is used to separate fringes from different wavelengths, the measured Doppler error per fringe becomes f o = (SX )1/2 f (2.12)

The total Doppler error over a wave band of AX can be approximately described as ORV « SXiS-0-5AX-0-5R-0-5D-1. (2.13)

Pixel Nnmbti

Fig. 2.3. An example of a fringing spectrum. The dashed line is the raw fringe. The cross points are the fringe after flat fielding and normalization, which can be fit with a sinusoidal function (the solid line) to extract the fringe phase information. The Doppler shift, AV is proportional to the phase shift, A0.

Pixel Nnmbti

Fig. 2.3. An example of a fringing spectrum. The dashed line is the raw fringe. The cross points are the fringe after flat fielding and normalization, which can be fit with a sinusoidal function (the solid line) to extract the fringe phase information. The Doppler shift, AV is proportional to the phase shift, A0.

This formula resembles that for the echelle (Eq. 2.5); the main difference is the dependence on the instrumental resolution (1/2 power for DFDI; 3/2 power for echelle). This allows DFDI instruments to use a medium resolution, high efficiency, first-order grating spectrometer for dispersing the fringes, producing a dramatically reduced size (and cost) of the instrument, while maintaining high precision for RV measurements. The fringing data for a single star requires only a small area in the detector plane for recording. This latter property enables simultaneous RV measurements of many objects using a reasonably sized detector (Ge 2002). The high throughput and multi-object capability are the main advantages of DFDI compared to the single-object echelle approach. The high throughput gain can offset the Doppler sensitivity loss due to the use of a much lower dispersion grating for a DFDI instrument than for an echelle instrument.

The simple instrument response, the sinusoidal function, created by the two-beam interference allows the DFDI method to be easily adopted to other wavelengths for maximizing the Doppler detection sensitivity for different stellar spectral types. The wavelengths other than the optical region include near UV and blue wavelength region (380-500 nm), red region (700-1000 nm) and near IR region (1.01.8 pm). Therefore, the DFDI survey instruments can be designed to target stars from late F type in the near UV to optical to late M types in the near IR. On the other hand, although the multi-object DFDI instrument interferometer is usually coupled with a first order grating spectrometer, it can also be designed to couple with a high efficiency cross-dispersed high resolution echellette or echelle spectro-graph to gain additional Doppler sensivity by increasing the operation wavelength coverage and dispersion power. Since the Doppler sensitivity weakly depends on the spectrograph resolution, the spectrograph can still be designed to have moderate to high resolution (such as R^ 20,000) to reach high Doppler sensitivity. This kind of design can still leave sufficient detector resources to pack fringing spectra from tens of objects on a large size CCD detector (such as 4kx4k CCD) to allow multi-object high precision RV measurements.

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