Erratum: ``Linear and Bayesian Planet Detection Algorithms for the Terrestrial Planet Finder'' (ApJ, 646, 1260 [2006])

Details Erratum: ``Linear and Bayesian Planet Detection Algorithms for the Terrestrial Planet Finder'' (ApJ, 646, 1260 [2006]) Erratum: ``Linear and Bayesian Planet Detection Algorithms for the Terrestrial Planet Finder'' (ApJ, 646, 1260 [2006])

Jan 2008

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008apj...672..734k&link_type=abstract

The Astrophysical Journal, Volume 672, Issue 1, pp. 734-734.

Mathematics

Probability

Scientific paper

We have discovered a slight error in the formulas for integration time to achieve a detection using either matched filtering or Bayesian hypothesis testing, due to our treatment of the Airy throughput. While the basic formulation is correct, and the dependencies on the probability of false alarms and missed detections are correct, the error in the constant β would cause a precise numerical calculation of the integration time to be in error.
In equation (17) we gave an expression for the electron rate from the detector due to a planet and background,Îij(t)=ɛη2ΔλIp(λr)Δα¯(A2c)/AsP¯ij+ɛη2ΔλIbΔα. (17)
We then substituted for the total throughput. This substitution, T=Ac/A, was only valid for binary apodizations or open apertures (such as a Lyot stop); that was not clearly stated. For a general apodized aperture the total throughput is reduced,T=(Ac)/A[1/(Ac)SA2(x,y)dxdy.
Letting τA equal the dimensionless quantity in brackets, equation (18) for the electron count rate then becomesÎij(t)=ɛη2ΔλIp(λr)Δα¯(T/τA)2AsP¯ij+ɛη2ΔλIbΔα. (18)Note that τA is 1 for an open aperture or shaped pupil, leaving the equation as given in the paper.
The error occurs in our expression for β in equation (33). Finding equation (33) from equation (34) involved, among other steps, replacing the total throughput by the Airy throughput. The expression for Airy throughput below equation (33) is incorrect. The Airy throughput, that is, the amount of energy in the central core of the PSF as a fraction of the total energy, is correctly given byTA≡(A2c)/(A2)sΔSP¯(ξ,ζ)dξdζ=(T2)/(τ2A)sΔSP¯(ξ,ζ)dξdζ.
Using this relationship between the total throughput and the Airy throughput, involving the sum of a region ΔS around the center of the PSF, equation (33) in the paper is still correct,t=1/β((K-γsqrt(1+Q˜Ξ/Ψ))2)/(Q˜TAΨ), (33)but with β=ɛη2ΔλIp(λr)A.
Likewise, the expression for the integration time with Bayesian hypothesis testing in equation (64) is still correct, but with the corrected expression for β above.
Finally, because the incorrect value for β was used in § 4, the numerical results are slightly in error. The proper integration time in equation (39) for the parameter values given ist≅9((4+3.1sqrt(1+Q˜Ξ/Ψ))2)/(Q˜TAΨ), (39)which uses a value of β=0.11 rather than 0.06. The resulting integration time is thus roughly a factor of 2 smaller, or 1 hour.

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