A&A 441, 1205-1210 (2005)
Laboratoire d'Astrophysique de Grenoble and Jean-Marie Mariotti Center, Université Joseph Fourier, 38041 Grenoble Cedex 9, France
Received 10 March 2005 / Accepted 2 May 2005
We investigate the possiblity to detect Earth-like planets, in the visible and the near infrared domains, with ground based Extremely Large Telescopes equipped with adaptive systems capable of providing high Strehl ratios. From a detailed analysis of the speckle noise, we derive analytical expressions of the signal to noise ratio on the planet flux, for direct and differential imaging, in the presence of the speckle noise and the photon noise of the residual stellar halo. We find that a 100 m telescope would detect an Earth at a distance of 10 pc, with a signal to noise ratio of 5, in an integration time of 12 h. This requires to control the instrumental aberrations with a precision better than 1 nanometer rms, and to reach an image dynamics of at radius. Under the same conditions, a telescope of 30 m would require a dynamics of for a positive detection.
Key words: instrumentation: adaptive optics - techniques: interferometric - techniques: high angular resolution - planetary systems
Since 1995, more than 130 planets have been found through radial velocity measurements surveys (Udry et al. 2003). The next important step forward will be the detection and characterization of exoplanets with ground based Extremely Large Telescopes (Angel 2003). The aim of the present work is twofold: (1) establish analytic expressions of the signal to noise ratio on the planet flux in the presence of speckle and photon noise, (2) study the possible detection of an Earth-like planet, in the visible and the near infrared domains, with an Extremely Large Telescope. We compute the spatial intensity distribution of the long exposure coronagraphic image in Sect. 2. In Sect. 3, we introduce the spatial and spectral covariance coefficient of the speckle noise. Analytical expressions of the signal to noise ratio (SNR) on the planet flux are derived in Sects. 4 and 5, for direct imaging and differential imaging. Finally, the results for an Earth-like planet are discussed in Sect. 6.
Let us consider an Extremely Large Telescope provided with a monolithic pupil of diameter D and equipped with an adaptive optics system capable of providing high Strehl ratios (>0.5). We observe in the focal plane of the telescope the image of an unresolved star through a -rotation shearing coronagraph (Gay et al. 1997; Roddier & Roddier 1997), whose effect is to reduce the image dynamics by cancelling the coherent part of the light. The imaging system, telescope included, is assumed to be free of instrumental aberrations.
At high Strehl ratios
the long exposure image of the telescope at the angular position
and at the
partially corrected by adaptive optics, can be written as (see Appendix A):
The incoherent part of the telescope point spread function, , is equal to the power spectrum of the residual phase over the telescope pupil (Rigaut et al. 1998). In the simple analytic model proposed by Jolissaint & Veran (2002), the latter is the sum of five terms. We keep the two dominant terms, i.e. the servo-lag spectrum and the uncorrected high frequency spectrum, assuming a noise free wavefront sensor, no aliasing (Verinaud et al. 2004) and no anisoplanatism. Figure 1 shows a radial cut of the coronagraphic images in the V (m), R (m), I (m), J (m) and H (m) photometric bands, with an actuator spacing of 20 cm, a seeing of at m, an outer scale of the atmosphere of 25 m, a wind speed of 10 m/s and time constants of 0.37 ms for the adaptive correction. The Strehl ratios are 0.67, 0.80, 0.88, 0.94 and 0.965, respectively.
To quantify the image quality, we introduce the dynamics of the image, , defined as the ratio of the intensity of the perfect stellar image in the center of the field to the intensity of the incoherent halo at the angular position . For a 100 m telescope, the dynamics at the distance varies from at m to at m. As the dynamics is proportional to D2, it is 10 times lower for a 30 m telescope.
The speckle noise is related to the intensity variations produced by the phase fluctuactions over the telescope pupil. In practice, the signal,
hence the speckle noise, are integrated over a field of view
and a spectral bandwidth
In order to
make a proper derivation of the speckle noise, we need to compute its spatial and spectral covariance
In the limit of high Strehl ratios, it is (see Appendix B):
The variance of the speckle noise is equal to twice the square of the halo intensity. If we exclude the factor of two resulting from the addition of two incoherent speckle fields in the coronagraphic image, the same result has been established by Dainty (1974) for pure turbulent images and extended later to the incoherent part of partially corrected images (Goodman 1985; Canales & Cagigal 1999; Aime & Soummer 2004). Hence, Eq. (3) is valid at any Strehl ratios. The correlation coefficient of the speckle noise is equal to the Airy disk function .
|Figure 1: Intensity at the output of the coronagraph at 0.55, 0.70, 0.90, 1.25 and m, normalized to the maximum of the perfect point spread function. The curves represent the inverse of the image dynamics as defined in Sect. 2 (D=100 m; Fried parameter: ; actuators spacing: 20 cm; outer scale of the atmosphere: 25 m; wind speed: 10 m/s; time constants: 0.37 ms). The Strehl ratios are 0.67, 0.80, 0.88, 0.94 and 0.965, respectively.|
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We concentrate on a system formed by an unresolved star in the center of the field and a planet located at a distance
than a few Airy disks. The image of the planet appears as 2 symmetrical contributions superposed on the stellar halo.
Adding the two symmetrical contributions of the planet provides a useful signal
where f is the flux ratio between the planet and the star.
is given by:
|Figure 2: Ratio , normalized for a zero field, as a function of the integration field for monochromatic (full line) and broad band (broken line, ) images. Note that this ratio is independent of the telescope diameter, and beyond a few Airy disks distance, its shape is also independent of .|
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Figure 2 shows the variation of the ratio
as a function of the integration field in Airy disk units,
for monochromatic and low spectral
resolution images (
). For broad bands, the loss of SNR is only
for a field of view per pixel
is maximum in the limit of a zero field, and can be approximated within a few
To optimize the SNR, we will use this maximum value and the
(Naylor 1998). Then, introducing the dynamics of the image
at the position of the planet, the SNR becomes:
The parameter p can be written as
is the angular position
of the planet expressed in Airy disk units. For broad band images, p is usually smaller than 1, and the quantity
be approximated by
Hence, the flux
of the star
- for which the speckle noise is equal to the photon noise is given by:
Differential imaging consists in making the difference between two coronagraphic images at different wavelengths in order to decrease
the speckle noise at levels lower than the photon noise. We compute the difference between two images at close wavelengths
scaled at the wavelength
weighted with factors
the difference between the two speckle patterns. At high Strehl ratios, this difference is given by
is the distance between the two bands (Marois et al. 2000).
Better weights could be found if we knew the exact Strehl ratio - see Eq. (3) -
but it is a priori not the case, and in practice they have to be estimated experimentally. Assuming for simplicity the
same spectral resolution
for the two bands, the measured differential signal
from the planet,
obtained as above by adding the two symmetrical contributions, is
is the planet relative flux fraction between the two bands. Let
be the differential
speckle noise from the stellar halo, the variance of
is given by
The scaling of the images allows optimizing the relative flux fraction .
Indeed after scaling, the planet contributions
in the image difference are shifted apart from each other by the quantity
or in other words, if
they are fully separated and
Optimum observing conditions are reached if, for any spectral channel, it is possible to find at least another spectral channel
at a distance n, large enough for the previous condition to hold, and small enough for the
differential speckle noise to remain smaller than the photon noise. This is the case of an Earth at a distance larger
than a few parsec if the total optical bandwidth is large enough. Under these conditions, the photon noise dominates and
the SNR per spectral channel can be written as:
|Figure 3: Signal to noise ratios on the planet flux as a function of the distance, for imaging ( a) and b): ) and full contrast differential imaging ( c) and d): , a spectral resolution of 100 and the combined signal of 20 bands), for an integration time of 12 h and telescope sizes of 30 and 100 m, respectively (the dotted lines correspond to a SNR of 5).We adopt a constant Earth to Sun flux ratio of and Sun fluxes at 10 pc of 45, 54, 55, 54, and 47 Jy in the V, R, I, J and H photometric bands, respectively. For the calculations we used the coronagraphic images of Fig. 1, a speckle lifetime of , an effective aperture of 0.85D, and an instrumental efficiency t=0.2. The SNR's at 10 pc are 0.4 and 3 for a 30 m telescope and, 8 and 30 for a 100 m telescope, with imaging and differential imaging, respectively.|
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We investigate the possibility to detect planets like the Earth, orbiting a star like the Sun. For this purpose we adopt a constant Earth to Sun flux ratio of and Sun fluxes at 10 pc of 45, 54, 55, 54, and 47 Jy in the V, R, I, J and H photometric bands, respectively (Allen 1977; Wamstecker 1980; Des Marais et al. 2002). The SNR's on the planet flux are plotted in Fig. 3 as a function of the distance, for an integration time of 12 h and telescope diameters of 30 and 100 m. Figures 3a and 3b represent the SNR for direct imaging with broad band images ( ), and Figs. 3c and 3d represent the SNR for full contrast ( ) differential imaging, with a spectral resolution of 100 and the combined signal of 20 couples of adjacent bands. The dynamics required to reach a SNR of 5 is plotted in Fig. 4 as a function of the distance, for each observing condition. Within a factor smaller than 2, all wavelengths provide the same SNR. In the following, we discuss the detection of an Earth at a distance of 10 pc.
With direct imaging, the speckle noise dominates and shorter wavelengths (V, R) provide slightly better results.
A 30 m telescope would give a SNR of about 0.4, which would not allow to detect the Earth. However,
a 100 m telescope would detect the Earth with a SNR of 8. Differential imaging provide better SNR's. In this case
the photon noise dominates and the I, J and H bands are slightly more favourable. The SNR's achieved are 3 and 30 for telescopes diameters of 30 and 100 m, respectively, allowing a marginal and a clear detection. For a 30 m telescope, achieving a SNR of 5 would require increasing the dynamics
by a factor 3 at m to reach
at m). For a 100 m telescope, the dynamics required for a positive
detection would be 10 times smaller,
at m (
at m). At the present time the best adaptive systems provide at most a
dynamics at m of
radius (Codona & Angel 2004). Clever methods have been proposed
to reduce the residual speckle noise (Codona & Angel 2004; Guyon 2004; Labeyrie 2004) but their real performances have to be evaluated.
Hence, to achieve a SNR of 5 on the flux of an Earth-like planet at a distance of 10 pc with a 100 m (resp. 30 m) telescope,
the first challenge is to produce images with a dynamics at
radius a few hundred times larger (resp. a few thousand) than has been presently
radius. Achieving a SNR of 5 per spectral band at a resolution of 100 would require a dynamics 20 times larger
(see Fig. 5).
|Figure 4: Image dynamics required to achieve a SNR of 5 as a function of the distance for imaging ( a) and b)) and differential imaging ( c) and d)) - see Fig. 3 for the experimental parameters. In the case of differential imaging, beyond a spectral resolution of a few tens the dynamics required is independent of the spectral resolution. At a distance of 10 pc, the dynamics required to achieve a SNR of 5 (dotted lines) are and for telescope diameters of 100 m and 30 m, respectively.|
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Up to now, we have assumed a perfect optical system. In the case of real systems, especially with segmented pupils, instrumental
aberrations will induce a noise similar to the speckle noise, given by:
|Figure 5: Image dynamics required to achieve at 10 pc a SNR of 5 per spectral channel as a function of the spectral resolution (differential imaging, see Fig. 3 for the experimental parameters). For a spectral resolution of 100, the dynamics required (dotted lines) are and for telescope diameters of 100 m and 30 m, respectively. The points at a spectral resolution of 5 correspond to the dynamics required with broad band imaging.|
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The author would like to thank the referee Dr. R. Racine for his helpful comments that deeply improved the clarity of the paper, and Drs M. Swain, E. Tatulli, C. Verinaud, H. Zinnecker and the Opticon ELT working group on exoplanets for helpful discussions.
The long exposure image of the telescope, at the angular position
and at the wavelength ,
partially corrected by adaptive optics, is given by:
The effect of the rotation shearing coronagraph is to split the pupil into two parts, to rotate one of the pupils by
to shift the phase of the other pupil by ,
and then to add the two pupils. The complex amplitude
in the exit pupil at the output of the coronagraph is given by:
The spatial and spectral covariance of the speckle noise is given by: