The two key performance models are described in this section: the Interferometer Performance Model (IPM) for predicting the SNR on a specified target; and the Star Count Model for estimating the number of targets that can be surveyed over the duration of the mission.

Interferometer Performance Model

Figure 1. Simplified TPF-I error budget for the signal-to-noise ratio (SNR) required for ozone detection at 10-µm wavelength.

The TPF Interferometer Performance Model (IPM) is maintained as a series of Excel spreadsheets and is used both to estimate the performance of different architectures, and to derive the requirements for the baseline design. The spreadsheet performance model is run as a bottom-up calculation; that is, the overall performance is calculated based on the specified low-level inputs (as opposed to a top-down suballocation approach). The inputs can be adjusted and balanced until the desired output performance is obtained, at which point the input values assumed become the requirements on the instrument.

Figure 1 shows a simplified version of the TPF-I error budget. The values shown result in an SNR of 10 for an ozone spectral channel spanning 9.5 to 10.0 m, which currently represents the driving requirement on the integration time. The Earth-like planet orbits a G2 Sun-like star with an angular separation of 50 mas at 15 pc distance. The ecliptic latitude is 30 degrees (this determines the contribution from local zodiacal dust). The array has a linear Dual Chopped Bracewell configuration, comprising four 4-m diameter collectors spaced at 30-m intervals with phases of 0, π, π/2, and 3π/2, for a total array size of 90 m.

The overall SNR is built up from 53 rotations of the array, each with a period of 50,000 s. The total observing time is 31 days (which does not currently include any overhead for calibration). The SNR for a single rotation is broken out into the root-mean-square (rms) of the planet signal variations and the contributions from random and systematic noise. The rms planet signal in this spectral band is less than 0.1 photons/s, corresponding to a fraction 2.3 x 10^-8 of the stellar signal in the same channel. In this example, the random noise has approximately equal contributions from the stellar size leakage and from local zodiacal dust. Other contributions (including exozodiacal dust emission, instrument thermal emission and stray light) have been omitted for clarity. The conversion from leakage photon rate to leakage photon noise is based on shot noise for a rotation of 50,000 s. The null floor leak term represents the photon noise arising from mismatches in the instrument beamtrains. The null floor makes a much lower contribution to the random noise budget than the local zodiacal emission and stellar leakage since it is driven by need to minimize the systematic error.

The instability noise contribution to the error budget is indicated by the blue boxes, and it is chosen to be similar in magnitude to the random error. The contribution of 0.051 photon/s corresponds to a null fluctuation of order 10^-8 at frequencies similar to the planet signal. These null fluctuations result primarily from nonlinear combinations of amplitude (ΔA) and phase (Δø) errors of the electric fields from the collectors. Analysis shows that the electric fields delivered by each collector must be matched in amplitude to within an rms error of less than 0.13% (equivalent to 0.26% intensity error), and matched in phase to within 1 milliradian at λ = 10 µm (equivalent to 1.5 nm of path). These conditions must be met simultaneously for all wavelengths in the science band, for both polarization states, and over all timescales (including DC offsets and vibrations in the kilohertz frequency range). The null depth resulting from this level of control is 7.5 x 10^-7. Meeting these amplitude and phase requirements is the primary technical challenge for the TPF-I system, and these requirements drive almost all aspects of the instrument design. Instability noise and its mitigation are described in more detail on the page devoted to Instability Noise and the X-Array.

In Figure 1 the amplitude and phase errors have been further categorized into static and dynamic terms. Static errors arise from mismatches in the coatings, the reflective and transmissive optics, and the static alignment of the system, including both dispersive and birefringent effects. Introducing an achromatic phase shift in the nulling beam combiner has been a focus of research, but matching the transmission of the different beamtrains - each of which contains on the order of 30 optical elements - across the full range of wavelengths and polarization is also a formidable challenge. The dynamic terms include all time-variable effects. The formation-flying system is continually in motion, and a series of control systems must be used to stabilize the optical path at the 1-nm level and to manage the tilt and shear of the wavefront that couples into the single-mode spatial filter. It is clearly important to validate the static terms over an optical bandwidth that is representative of the flight system. Conversely, if the instrument can be demonstrated to be stable at one wavelength, then we can argue that it will be stable at all wavelengths (with the exception of time-varying dispersive terms, which should be small).

Figure 2. Integration times.

The actual error budget is somewhat more detailed than the one shown previously in Figure 1. The SNR is calculated over a number of spectral channels spanning the full bandwidth of the instrument. The model is flexible and can compute the SNR for arbitrary array geometries, with and without phase chopping, using the method of baseline decomposition (see Lay 2004). Noise sources include photon noise from local zodiacal emission, stellar leakage, exozodiacal emission, instrument thermal emission, stray-light, detector read noise, and dark current, as well as a sophisticated calculation of the instability noise. The contributions from local and exozodiacal dust are based on the models of Kelsall et al. (1998). The signal and noise sources have different dependences on wavelength, distance, array size, and collector diameter, leading to complex variation in the broad-band SNR. Zodiacal and exozodiacal emission started appearing.

Star Count Model

The purpose of the star count model is to predict the number of stars that can be surveyed for the presence of planets during a specified mission duration. Here we provide a brief description. A more detailed explanation can be found in Dubovitsky and Lay (2004).

Originally implemented in MathCad, the model has now been migrated to MatLab. The starting point is the catalog of 1014 TPF candidate target stars described previously and illustrated by the circles in Figure 2. Given a specific nulling configuration and collector diameter, the program cycles through each star, determining first whether it meets the criteria for ecliptic latitude (solar shading constraint) and inner working angle. It then estimates the time needed to achieve an SNR of 5 for a broad-band detection of an Earth-sized planet, using a simplified version of the full performance model. The list of targets is then sorted in ascending order of integration time. Following the Science Requirements available for the initial survey, we assume that 50% of this time will be spent integrating on targets and that each target will be visited three times to ensure a completeness in excess of 90%. The number of targets that can be surveyed is therefore given by the star in the sorted list by which the cumulative integration time has reached 2 x 50% / 3 = 122 days.

The colored circles in Figure 2 indicate the integration time needed to detect an Earth-sized planet around the star. The mid-IR signal from an Earth-sized planet in the mid-Habitable Zone depends only on the distance to the system. The noise is dominated by contributions from the local zodiacal dust (invariant with distance) and stellar size leakage (which decreases with distance but increases for the earlier spectral types). As a result the easiest targets in red are the nearby K stars. Integration times increase with distance and with the intrinsic size of the star. The solid line shows schematically a contour of constant integration time; targets to the left are limited by local zodiacal emission, while those to the right are limited by stellar leakage.

This technique can also be extended to predict the number of targets that can be spectroscopically characterized in a given amount of time. This depends on the prevalence of Earth-like planets in the Habitable Zone. If planets are rare then the average distance to the systems being characterized is high, the integration times are long, and only a few systems can be accommodated in the time available. If planets are very common, however, then we will be able to characterize a much larger number of nearby systems. The simple algorithm to account for this is described in Dubovitsky and Lay (2004).

A more sophisticated model to predict and optimize the program completeness is now underway. Based on similar analysis for the TPF-C mission, the algorithm assesses the observability for each of 1000 planets around each star. For each week of the mission, only the most productive stars are selected in order to maximize the number of planets found.