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KSFGround-based optical interferometry
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Ground-based optical interferometry, as reviewed here, is the application of long-baseline optical interferometric techniques to astrometric measurements over a wide range of angular scales, extending from cataloging star positions over the entire sky down to the study of binary stars at angular separations ≤ 1 milliarcsecond (1 mas; Zacharias 2010). Large-scale optical measurements are made for the purpose of setting up and maintaining a celestial, global coordinate frame in the optical and determining positions and motions within it. Small-angle measurements include those made for the determination of binary star orbits and masses, trigonometric parallaxes, proper motions, the astrometric detection of extra-solar planets, and in general, the determination of the relative positions of celestial objects for whatever purpose.
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[edit] Introduction
Classical optical astrometry used the meridian transit circle to determine the position of stars over large angles to accuracies of order 0.05 - 0.1 arcseconds (Hughes & Hutter 1990). By comparison, modern space missions utilizing drift-scanning techniques are capable of far higher accuracy. For example, the Hipparcos satellite produced a reference frame of stars with accuracies of ~ 1 mas at the epoch of the observations (c. 1991, Perryman et al. 1997). However, Hipparcos positional accuracies have degraded over the intervening years to ~ 20 mas (c. 2011) due to proper motion uncertainties. As space techniques are very costly, ground-based optical interferometers play an important role. With operational lifetimes measured in decades, they have the potential to maintain the accuracy of these catalogs by improved measurements of the proper motions of these and additional stars.
The classical technique for narrow angle astrometry was photographic astrometry by which stellar positions were determined from the precise measurement of the stars' images on photographic plates. By averaging thousands of measurements, a formal error of ~ 1 mas was achieved in the relative positions of stars over fields of a few arc minutes. Modern techniques utilizing CCD detectors achieve precisions of ~0.5 mas with relatively fewer measurements. At even smaller angular scales, speckle interferometry (Labeyrie 1970) provides the ability to reach the diffraction limit of a telescope (~ 30 mas on a 4 meter telescope), and has provided the possibility of significant improvement of precision of measurement, yielding typical errors of 0.5% in separation (Mason et al. 1999) for a 2-3 arcsecond field of view. This field of view is limited by atmospheric turbulence and thus this technique has been limited to close double stars. By comparison, long baseline optical interferometry can potentially achieve orders of magnitude improvement in accuracies, down to levels of microarcseconds (μas) for fields ≤ 30 arcseconds.
[edit] Optical interferometry basics
Optical interferometry, like radio interferometry, images synthetically through observation of an object’s complex fringe visibility at various baseline lengths and orientations. These observations are related to the object’s irradiance distribution through the van Cittert-Zernike theorem (Thompson et al. 2001), which gives the irradiance distribution as a Fourier transform of the complex visibility measurements \( \mathit{Ve^{iφ} }\) (where V and φ are the fringe amplitude and phase). Powerful inversion algorithms (Pearson 1999) have been developed to transform visibility measurements to the object irradiance (Sec. 5.2: Binary stars - Current status).
While the fringe phase φ is necessary for true imaging, measurements of just the fringe amplitude can be used to fit models of simple objects---disks, dust shells, binary stars (Sec. 5.1: Binary stars - Observations and analysis), etc. Conversely, for most astrometric applications (Sec. 3: Wide-angle astrometry & Sec. 4: Narrow-angle astrometry) measurements of the fringe phase are of primary interest. Figure 1 shows a schematic of a simple two-element interferometer. Light is collected by the apertures and is routed to a combining beamsplitter. Variable optical delay lines (Figure 2) in the arms of the interferometer match the path delays from the incident wavefront to the beam combiner. If these pathlengths are matched to within a few wavelengths, an interference fringe is detected. The complex fringe visibility is the phase and amplitude of the interference pattern. The delay is the measured difference of the delay-line positions at which the fringe is detected.
[edit] Wide-angle astrometry
The goal of wide-angle astrometry is to determine accurate positions of stars over the entire sky. The problem of astrometry via interferometry is to recover the two coordinates for each star from the observed delays. In the absence of atmospheric effects, the geometrical delay can be defined as (Armstrong et al. 1998) \[\tag{1} d_{G, ij}(t) ≡ d_{j}(t) - d_{i}(t) = \mathbf{B}_{ij}(t) \mathbf{ ⋅ \ ŝ}_{0} - C_{ij},\] where the geometrical delay is the difference between the delay line positions \(\mathit{d_j}\) and \(\mathit{d_i}\) that is required to equalize the effective optical paths from the star to the point of beam combination via each of two apertures i and j, \(\mathbf{B}_{ij}(t)\) is the baseline vector between the apertures, \(\mathbf{ŝ}_{0}\) is the unit vector toward the star position, and \(\mathit{C_{ij}}\) is the difference between the “fixed” internal optical path lengths \(\mathit{C_i}\) and \(\mathit{C_j}\) within the instrument. In principle, sufficient delay measurements would allow the determination of the interferometer baseline vectors and delay constants, along with the positions of the stars. However, the situation is greatly complicated by the atmosphere and the fact that neither the delay “constants” (\(\mathit{C_{ij}}\)) nor the baseline vectors are stable over time: The atmosphere induces large (≥ 1 µm) delay fluctuations on timescales as short as a few milliseconds. Thermal drifts in the positions of the apertures (typically siderostats, Figure 3) and all subsequent elements in the optical paths prior to the point of beam combination typically produce drifts of up to tens of µm/hour in the delay constants. Finally, mechanical imperfections in the mounts of the collecting apertures can produce ~ 10 µm changes in the baseline vectors each time the interferometer is pointed at a new star! The design of, and the analysis of the data from, any ground-based optical interferometer must overcome all of these effects.
[edit] Atmospheric corrections
The correction of the delay variations caused by the atmosphere relies on the fact that, in the optical, atmospheric dispersion varies in a significantly nonlinear manner with wavelength (Edlen 1966). The path change in the atmosphere above any array element i is equivalent, to good approximation, to replacing a length \(P_{\mathit{i}}\) with air. The change in equivalent path length is given by \(A_{\mathit{i}}(σ, t) = P_{\mathit{i}}(t)[n(σ) – 1],\) where n(σ) is the refractive index of air and σ is wavenumber (1/wavelength). The observed delay is then given by \[\tag{2} d_{ij}(σ, t) = \mathbf{B}_{ij}(t) \mathbf{ ⋅ \ ŝ}_{0} - C_{ij} – A_{\mathit{ij}}(σ, t) = d_{G,ij}(t) + d_{A,ij}(σ,t),\] where \(A_{ij} ≡ A_{j} - A_{i},\) and the atmospheric delay \(d_{A,ij}\) is the delay required to compensate for \(A_{ij}\).
The observed phase \(φ_{ij}\), measured with respect to the direction \(\mathbf{ŝ}_{0}\) is then given by \[\tag{3} φ_{ij}(σ, t) = φ_{S,ij}(σ) + φ_{A,ij}(σ, t),\] where \(φ_{S,ij}(σ)\) is the (vacuum) source phase, and the atmospheric phase \(φ_{A,ij}\) is defined by \[\tag{4} φ_{A,ij}(σ, t) ≡ 2πσd_{A,ij}(σ,t) = -2πσ[A_{j}(σ, t) - A_{i}(σ, t)].\] The variation of \(φ_{A,ij}(σ, t)\) with σ can then be used to estimate \(A_{ij}\), because \(n(σ) – 1\) has a significant \(σ^2\) dependence at optical wavelengths. Expanding \(φ_{A,ij}\) as a Taylor series around some \(σ_0\) yields \[\tag{5} φ_{A}(σ) = φ_{A}(σ_{0})[D_1 + σ D_2 + Φ(σ)],\] where the D coefficients depend on \(σ_0\), \(n(σ_{0}) – 1\), dn/dσ, and \(d^{2}n/dσ^2\). The first term gives a constant offset in phase. The second term produces a phase term that is indistinguishable from an error in the position of the star (by the Fourier transform shift theorem, Bracewell 2000). The Φ(σ) term makes it possible to determine the air-path mismatch. With the air-path mismatch determined, we can calculate \(d_{A,ij}(σ)\) and subtract it from \(d_{ij}(σ)\), resulting in an estimate of \(d_{G,ij}\), the desired datum for determining the stellar positions (eq. (1)).
[edit] “Constant term” corrections
Several techniques can be used to measure the temporal variations in the delay zero-point (\(C_{ij}\)) on each interferometer baseline. Since rapid path length variations are typically present, an internal metrology system must be used to continuously monitor the optical paths through the instrument. A single-color laser metrology system can be employed to measure the relative changes in the paths, in combination with occasional fringe tracking observations of an internal white-light source, with the siderostats in autocollimation, to measure the absolute differential path length. Alternatively, a two-color absolute metrology system could be employed (Peters et al. 2004).
[edit] Baseline metrology
Since the astrometric errors introduced by baseline instability scale as the fractional accuracy in the measurement of the baseline (δB/B), 1 mas astrometry requires knowledge of a 20 m baseline to 100 nm. As noted above, the baselines are not stable to that degree, due to instability of the siderostat mounts. However, since it can be shown (Hines et al. 1990) that any point on the siderostat mirror surface can, if consistently used, be defined as one end of the baseline vector, the motion of a retroreflector on the mirror surface can be monitored to determine the temporal variation of the baseline vector. Such measurements can then be used to correct for the temporal variations in the baseline in the astrometric solutions (eq. (1)).
[edit] Current status
Currently, the Navy Optical Interferometer (NOI, formerly NPOI, Armstrong et al. 1998, Figure 4) is the only long-base optical interferometer specifically designed to measure and compensate for the effects of the atmosphere and the temporal instability of the delay “constants” (\(C_{ij}\)) and the baseline vectors. The use of vacuum delay lines renders the interferometer insensitive to atmospheric refraction and allows simultaneous fringe tracking in spectral channels over a wide bandpass (550-850nm). The delay on each baseline is modulated over a 2 ms period by a small number of wavelengths while, for each spectral channel and baseline, synchronously measuring the photon count rates with the phase of the delay modulation. The complex Fourier transform over all the channels yields the “group” delay (Lawson 1995), knowledge of which for each 2-ms period allows one to rotate the complex visibility phasors by \(e^{2π\mathit{i}dσ}\), where d is the group delay and σ is the wavenumber of the channel. This allows the coherent addition of the data to provide sufficient signal-to-noise to determine the variation of the fringe phase with wavenumber and thus determine the dispersion correction.
The NOI utilizes single-color laser metrology, as described in Sec. 3.2 ("Constant term" corrections), to measure the variations in the \(C_{ij}\) terms. The time-variations of the baselines are measured using a metrology system (Hutter & Elias 2003) consisting of a number of laser interferometers tied to reference tables, one next to each siderostat. Several interferometers measure the positions of a retroreflector near the intersection of the rotation axes of each siderostat relative to the table. The translation and tilt of the table is in turn measured by additional interferometers monitoring the table’s motion with respect to retroreflectors embedded in the bedrock beneath the observatory. The implementation of atmosphere and constant-term corrections at the NOI has resulted in positions of ~ 115 bright stars (Declinations ≥ -20°) with a median precision of ~8 mas (Figure 5, Benson et al. 2012). With the application of baseline metrology and other data analysis improvements, accuracies of this order are expected for these and many more stars. Radio star observations are being used to orient the preliminary catalog with respect to the fundamental reference defined by extragalactic radio sources. The longer-term goal of wide-angle astrometric observations with the NOI is to produce a catalog of positions for several thousands of the brighter Hipparcos stars with an internal accuracy of a few mas.
[edit] Narrow-angle astrometry
While atmospheric turbulence significantly limits the accuracy of wide-angle astrometric measurements, the limitations are far less severe for narrow-angle measurements. Measurements made in this regime are useful for a number of investigations, including the search for extrasolar planets. Astrometry measures the reflex motion of the parent star in the plane of the sky for evidence of an unseen companion and is complementary to radial velocity (spectroscopic) and other astrometric measurements. Interferometric observations of stars with planets known from the radial velocity surveys can yield the inclination i of the orbit and therefore the planet’s mass m, whereas only m sini can be determined spectroscopically.
[edit] Atmospheric effects
In conventional narrow-angle astrometry using single-aperture telescopes, the position of a target star is measured relative to a number of reference stars within the detector field of view. A detailed analysis using models of atmospheric turbulence (Lindegren 1980), as well as empirical data (Han 1989), shows that the turbulence-limited accuracy is proportional to the star separation θ to the 1/3 power, and to first order, is independent of the telescope aperture. The differential motion of stars attributable to turbulence occurs because the beams of light from each star follow different paths through the atmosphere (Figure 6, Shao & Colavita 1992). In conventional astrometry with long-focus telescopes, the size of the telescope D is much smaller than the separation of the beams high in the atmosphere (Figure 6, left), and the accuracy is independent of the diameter of the telescope. However, with a long-baseline interferometer, it is possible to construct baselines that are larger than the typical beam separation (Figure 6, right). In this regime, there is a qualitative change in the behavior of the atmospheric errors, resulting in strong dependencies on both baseline length and star separation. Accuracy now goes linearly with star separation, and improves with baseline length B to the -2/3 power. Adopting a particular atmospheric model (Shao & Colavita 1992) the error behavior becomes \[\tag{6} σ_{δθ} = 300B^{-2/3}θt^{-1/2},\] where \(σ_{δθ}\) is in arcseconds, B is in meters, θ is in radians, and t is the integration time in seconds. For a 20-arcsecond star separation and a 100-m interferometer, the atmospheric error in one hour of integration time should be less than about 20 μas, about two orders of magnitude improvement over conventional measurement techniques.
[edit] Other error sources
The measured delay can be written \[\tag{7} \mathit{x} = \mathit{l} + \mathit{k}^{-1}φ,\] where l is the laser-monitored internal delay, φ is the fringe phase, and k is the wavenumber of the interfering light. In the simplified case of two dimensions, and with the source near normal to the baseline so \(θ ≈ \mathit{x}/B\), one can show the error in the astrometric measurement to be \[\tag{8} δθ = \frac{δ\mathit{l}} {B} + \mathit{k}^{-1}\frac{δφ} {B} – \frac{δB} {B}θ.\] The first term incorporates the errors in measuring the internal delay, the second term incorporates errors in measuring the fringe phase, and the third term incorporates errors in measurement or knowledge of the interferometer baseline. Note that in the first two terms the longer baselines practical for ground based instruments directly contribute to reducing the astrometric error for a given measurement accuracy, while the dependence on θ in the third term illustrates the difference in the requirements on baseline knowledge between wide- and narrow-angle interferometric astrometry. For wide-angle astrometry, where θ ≈ 1 rad, the astrometric accuracy is equal to the fractional accuracy in the measurement of the baseline, while for small fields, the requirement on baseline accuracy decreases. For example, in a narrow field of 20 arcseconds (\(10^{-4}\) rad), the requirements on the baseline knowledge are reduced by a factor of ten thousand compared with a wide-angle measurement.
[edit] Implementing a narrow-angle measurement
Exploiting the tens-of-microarcsecond astrometric accuracy possible with a ground-based narrow-angle astrometric measurement requires the ability to utilize nearby reference stars. One approach to this problem uses a long-baseline interferometer with dual beam trains (Figure 7, Colavita et al. 1999). The light at each aperture forms an image of the field containing the target and reference stars. The light from each star is then fed to its own interferometric beam combiner. These beam combiners are referenced with laser metrology to a common fiducial at each collector. These common fiducials tie together the two interferometers allowing fringe-tracking errors on the primary star to be fed forward to the secondary star beam combiner. The secondary interferometer, cophased by the primary interferometer, makes the actual astrometric measurement by switching between the secondary star and the primary star; the change in delay between the two stars is the astrometric observable.
[edit] Current Status
The Palomar Testbed Interferometer (PTI, Colavita et al. 1999) demonstrated an astrometric precision of 100 μas between moderately close (~30 arcsecond) pairs of bright stars (Lane et al. 2000). The PTI was also used to observe binary stars with separations < 1 arcsecond (Muterspaugh et al. 2010a) where astrometric precisions of 40-50 μas were achieved. At these closer separations, it is possible to operate in a simpler mode, without the dual-star feed, since both binary components are within the field of view of a single interferometric beam combiner (Lane and Muterspaugh 2004). In this mode, a portion of the incoming starlight is directed to a second fringe-tracking beam combiner that measures the phase difference between the fringe packets of the two stars by modulating the instrumental delay. Observations of 51 multiple star systems made in this mode, yielded results that included precision binary orbits and component masses, studies of the orbits and physical properties of stars in triple (Figure 8, Muterspaugh et al. 2010b) and quadruple star systems, and candidate substellar companions in six binaries. The successor to PTI is the PRIMA (Phase Referenced Imaging and Microarcsecond Astrometry) facility (Delplancke et al. 2008) of the European Southern Observatory (ESO) Very Large Telescope Interferometer (VLTI), which among several functions aims to perform differential narrow-angle astrometry with an accuracy of 10 μas. A primary astrometric project for PRIMA is the detection and characterization of extrasolar planets. Other astrometric applications include the determination of parallaxes and measuring the dynamics of objects near the galactic center.
[edit] Binary stars
For binary stars with separations < 1 arcsecond, accurate relative astrometry is also possible through interferometric techniques that don’t require measurement of the absolute fringe phase. With the high angular resolution available, optical interferometers can resolve many spectroscopic binaries, and provide observations that complement speckle interferometry and single telescope, direct imaging results for binary systems with significantly eccentric orbits. Long baseline interferometry can independently determine all the parameters describing a binary star orbit, as well as the angular diameters of the stars and their brightness ratio. In conjunction with spectroscopic and photometric measurements, the fundamental physical parameters of the system, including masses, luminosities, colors, and distance, can be determined. For binaries where the components can be resolved, linear radii and effective temperatures can also be determined.
[edit] Observations and analysis
As noted in Sec. 2 (Optical interferometry basics), measurements of just the fringe amplitude can be used to fit models of simple objects. For a binary star, even a single-baseline interferometer can produce very useful results. At a given wavelength λ, the fringe amplitudes of the individual components are given by \[\tag{9} V_{1,2} ≈ 1 - \frac{(πD_{1,2}B_{λ})^2} {8},\] (Armstrong et al. 1992), where \(D_{1,2}\) are the angular diameters (mas) of the individual stars at this wavelength and \(B_λ\) is the “projected” baseline length (component of the baseline vector in the plane perpendicular to the source direction) in wavelengths. The maximum and minimum squared fringe amplitudes of the binary are then \[\tag{10} V^{2}_{max, min} = \left(\frac{FV_1 \pm V_2} {F + 1} \right)^2,\] where \(F= S_{1}/S_{2}\) is the brightness ratio of the components. As the projected interferometer baseline varies with Earth rotation, the fringe amplitude fluctuates for narrow bandpasses as shown in Figure 9 (left, Armstrong et al., 1992). Observations over many nights may thus be fitted to an orbit for the binary system (Figure 9, right).
[edit] Current status
Most modern optical interferometers include three or more apertures and thus can obtain simultaneous measurements of complex fringe visibility (Sec. 2: Optical interferometry basics) on each of several baselines. However, measured phases in ground-based interferometry are severely disturbed by propagation effects in the turbulent atmosphere. Fortunately, because the phase noise is additive at each telescope and the measured phases are phase differences between telescopes, the phase noise cancels when the measured phases are added up along a closed loop of three or more telescopes. This sum, referred to as “closure phase” is free of atmospheric phase noise, and is equal to the intrinsic source closure phase. Many modern optical interferometers exploit closure phase measurements to produce images (Figure 10, left; Zavala et al. 2010). These images can provide initial estimates of the separation and relative orientation of the stars subsequently used in simultaneous fits to all the complex fringe visibility data to derive accurate orbit solutions (Figure 10, right).
[edit] Summary
Ground-based wide-angle optical interferometric astrometry is beginning to update the best currently available catalogs of bright star positions. With an expected operational lifetime measured in decades, an optical interferometer such as the NOI can significantly improve the measure proper motions of these and additional stars. Position measurement repeated at regular intervals will also allow unambiguous separation of binary motion from proper motion, an accomplishment that might be difficult to achieve from future space-based observations of limited duration.
In the area of narrow-angle observations, ground-based optical interferometry has already contributed to the determination of dozens of binary and multiple star orbits and stellar masses, and is beginning to contribute to the study of exoplanets. Future developments in this technology promise even higher accuracies: For example, the GRAVITY project is developing a four-way beam combination instrument for the ESO VLTI system (Gillessen et al. 2010). Its main operation mode will make use of all four 8m Unit Telescopes (UT) to measure astrometric distances between objects located within a 2” field-of-view. With the sensitivity of the UTs and the 10 μas astrometric precision, it will be able to perform studies such as those of motions to within a few times the event horizon size of the Galactic Center massive black hole and potentially test general relativity in its strong field limit.
The successes of ground-based interferometry have also pointed the way toward future applications of long-baseline interferometry in space-based astrometric missions. While no longer being actively pursued, the SIM Lite Astrometric Observatory (Marr-IV et al. 2008) was to have been the first spacecraft whose primary mission is to perform precision astrometry by long-baseline interferometry. However, all the technology milestones towards an interferometer capable of positional accuracies of 4 μas or better were met during that project’s development, which would enable a similar mission in the future to pursue the detection of planets of a few Earth masses around nearby stars, as well as unprecedented studies such as of the internal dynamics of our galaxy, the dynamics in our local group of galaxies, and calibration of the extragalactic distance scale.
[edit] References
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Armstrong, J. T. et al. (1998). Astrophysical Journal, 496, 550.
Benson, J. A. et al. (2012). Astronomical Journal, in preparation.
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[edit] Recommended reading/links
Center for High Angular Resolution Astronomy (CHARA): http://www.chara.gsu.edu/CHARA/
Infrared Spatial Interferometer (ISI): http://isi.ssl.berkeley.edu/
Keck Interferometer (KI): http://keck.jpl.nasa.gov/keck_index.cfm
Large Binocular Telescope Interferometer (LBTI): http://nexsci.caltech.edu/missions/LBTI/
Lawson, P. R., ed. (1997). Selected Papers on Long Baseline Stellar Interferometry (SPIE Milestone Series, Volume MS 139). Bellingham: SPIE.
Lawson, P. R., ed. (1999). Principles of Long Baseline Stellar Interferometry (JPL Publication 00-009 07/00). Pasadena: JPL.
Magdalena Ridge Observatory Interferometry (MROI): http://www.mro.nmt.edu/about-mro/interferometer-mroi/
Navy Optical Interferometer (NOI): http://www2.lowell.edu/npoi/
NOI bibliography: http://www.usno.navy.mil/USNO/library/library-collections/noi-bibliography
Optical Hawaiian Array for Nanoradian Astronomy (OHANA): http://www.cfht.hawaii.edu/~lai/ohana.html
Optical Long Baseline Interferometry News (OLBIN): http://olbin.jpl.nasa.gov/
Sydney University Stellar Interferometer (SUSI): http://www.physics.usyd.edu.au/ioa/Main/SUSI
Very Large Telescope Interferometer (VLTI): http://www.eso.org/sci/facilities/paranal/telescopes/vlti/index.html
[edit] See also
Zacharias, N. (2010). Astrometry. Scholarpedia, 5(2):5267.