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Figure 7.3. Transmission of Earth's atmosphere from ground to space (vertical path). The dotted line is calculated transmission from sea level in a nominal standard atmosphere while the solid line is from an altitude of 4,000 m in a midlatitude summer atmosphere. The transmission advantage of placing telescopes on mountain tops is clear. In addition, since most of the absorption features are due to water vapor, telescopes are preferentially located in the driest regions of the world. In the near-IR range the spectral regions where atmospheric absorption is minimum, known as spectral "windows", are commonly called by the letter indicated. Hence, for example, the 2.1-2.4 ^m window is commonly known as the K-band and so on.

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Figure 7.3. Transmission of Earth's atmosphere from ground to space (vertical path). The dotted line is calculated transmission from sea level in a nominal standard atmosphere while the solid line is from an altitude of 4,000 m in a midlatitude summer atmosphere. The transmission advantage of placing telescopes on mountain tops is clear. In addition, since most of the absorption features are due to water vapor, telescopes are preferentially located in the driest regions of the world. In the near-IR range the spectral regions where atmospheric absorption is minimum, known as spectral "windows", are commonly called by the letter indicated. Hence, for example, the 2.1-2.4 ^m window is commonly known as the K-band and so on.

1,100 cm-1) most of the absorption is again due to water vapor. At 9.6 ^m (1,040 cm-1) there appears a strong absorption band of ozone, while at 15 ^m (667 cm-1) the strong v2 absorption band of C02 appears. At even longer wavelengths, absorption is dominated by the rotational absorption lines of water vapor, which then tends to zero towards microwave and radio wavelengths. At millimeter wavelengths, in addition to water vapor, 02 is also strongly absorbing at 2.5 mm and 5 mm due to magnetic dipole absorptions. At longer wavelengths, the atmosphere is effective transparent up to wavelengths of approximately 30 m, where it then becomes opaque again due to ionospheric effects.

Clearly the absorption features of water vapor are a major problem for terrestrial observatories, but fortunately most of the water vapor is held in the lower, warmer levels in the atmosphere. Thus, by placing the telescope at higher altitudes, and/or in desert regions, the absorption of water vapor can be greatly diminished as can be seen in Figure 7.3. In addition, since the pressure of the atmosphere, and thus column abundance of overlying air falls exponentially with height, the absorption of the other gases such as 02 and C02 is also reduced, together with any absorption due to dust or haze. The regions of low atmospheric absorption between the main absorption bands are known as spectral "windows". In the near-IR range these spectral "windows" are often called by the designations shown in Figure 7.3. Hence, the 1.9-2.5 ^m window is often referred to as the K-band, and so on. Clearly, even on dry mountain tops the absorption of the terrestrial atmosphere means that several very interesting regions of the giant planet spectra are unobservable. In addition, where the atmosphere is partially clear, the absorption depends on the abundance of highly variable atmospheric constituents such as water vapor and dust. Hence, whenever the spectrum of a planet is recorded, the spectrum of a nearby standard reference star must also be recorded so that the terrestrial absorption may be determined and corrected for. This correction leads inexorably to additional errors in the final recorded spectra.

7.3.2 Angular resolution

The second historical problem with ground-based observations of the giant planets is the achievable angular resolution. Figure 7.4 compares the physical sizes of all the giant planets and the Earth. However, for ground-based telescopes and Earth-orbiting space telescopes, the apparent angular size of the planets decreases greatly as we go from Jupiter to Neptune, and Figure 7.5 compares the apparent sizes of the planets as they appear at opposition where, for reference, the apparent diameter of Jupiter is ~40" (i.e., 40arcsec). Clearly the apparent size of Neptune is very small, making it very difficult to discern variable cloud features, although the disk-averaged brightness is easier to determine and has been monitored from the ground for decades (e.g., Lockwood and Jerzykiewicz, 2006; Lockwood and Thompson, 2002). While the angular resolution of space-based telescopes depends on aperture and optical quality alone, the angular resolution of ground-based telescopes is severely limited by the turbulence of the overlying atmosphere. The strength of this turbulence depends on local atmospheric conditions and is particularly noticeable in winter giving rise to the twinkling of the stars. Typically the "seeing" is limited to approximately 1" and in

Figure 7.4. Comparative sizes of the giant planets (Jupiter, Saturn, Uranus, and Neptune) and the Earth.
Figure 7.5. Relative apparent sizes of the giant planets as seen at opposition from the Earth with a telescope of infinite resolution.

Figure 7.6, the images have been blurred to this approximate resolution. While considerable cloud detail can still be seen on Jupiter the effect on the other planets becomes increasingly severe as the apparent angular diameter decreases.

The blurring of ground-based telescope images caused by atmospheric turbulence arises due to refractive index variations in the column of air between the telescope and the object. Light from a distant source, such as a star, arrives at the top of the Earth's atmosphere effectively as a plane wave. However, after passing

Figure 7.6. Relative appearance of the giant planets as seen at opposition from the Earth with typical "seeing" of approximately 1 arcsec resolution.

Figure 7.6. Relative appearance of the giant planets as seen at opposition from the Earth with typical "seeing" of approximately 1 arcsec resolution.

through the Earth's atmosphere to reach the telescope, variations in temperature caused by turbulence introduce small variations in the refractive index of the air which introduce randomly changing phase variations that continuously distort the wavefront and make it impossible to form a diffraction-limited image. The typical correlation time of the distortions is of the order of a few milliseconds and the problem is most evident at visible wavelengths, becoming progressively smaller at longer wavelengths due to variations in the flatness of the wavefront becoming smaller and smaller compared with the wavelength of the light observed. This problem severely affected terrestrial astronomical observations for many years, but recently technology has developed to such an extent that new techniques have been developed that go a long way to negating this problem, as will now be described.

Adaptive optics

The most ambitious technique for correcting the problem of "seeing" is adaptive optics, which attempts to remove distortions from the wavefront before the image is formed. This is achieved by simultaneously observing either a bright star close to the target or, if no such star is available, observing a simulated star formed by light scattered back from sodium atoms in the upper atmosphere from a powerful laser situated at the telescope facility. The important thing is to observe the "star" and the target through as nearly the same column of air as possible. Light from the guide star is collimated by the telescope and the flatness of the wavefront sensed. Data from the wavefront sensor is then used, via a suitable control algorithm, to modify a corrector plate, which attempts to null atmospheric distortions before the image is formed. The wavefront sensor and corrector plate operate in a closed loop and when operating correctly can effectively fully flatten the wavefront from the guide star. An image simultaneously recorded of the nearby target will be similarly corrected and will thus have an angular resolution much closer to the diffraction limit of the telescope depending on the sensitivity and resolution of the wavefront sensor and corrector plate, the correlation time of the distortions, and the efficiency of the control algorithm. An example of the power of adaptive optics is demonstrated in Figure 7.7, which shows images of Uranus observed by the Keck Observatory with and without the adaptive optics system engaged. The difference is startling.

Complex adaptive optics systems use correcting plates that can fully flatten the wavefront, while the simplest use is a basic tip-tilt system: simply a flat mirror whose mean angle is continually adjusted to keep the centroid of the guide star fixed.

Speckle imaging

A simpler method of image correction is speckle imaging. Since the correlation time of variations is of the order of a few milliseconds, images recorded with shorter exposures will each have a constant, nonvarying distortion. If thousands of short-exposure images are recorded, they can be analyzed and then suitably averaged to reconstruct near diffraction-limited images. Obviously this technique will work best for bright images due to the low signal-to-noise ratio of short-exposure images, and

Photo credit: Heidi EL Hoirimel, Imke de Poter, Keck Qbservotory

Uranus on 9 July 2004 The Power of Keek's Adaptive Optics

AO System OFF AO System ON

Photo credit: Heidi EL Hoirimel, Imke de Poter, Keck Qbservotory

Uranus on 9 July 2004 The Power of Keek's Adaptive Optics

AO System OFF AO System ON

Figure 7.7. Images of Uranus observed with the Keck Observatory in ambient observing conditions (left-hand images) and with the adaptive optics system turned on (right-hand images). The improvement in spatial resolution given by the AO system is enormous. Courtesy of Keck Observatory, California Association for Research in Astronomy.

the technique has been successfully applied by amateur astronomers to Jupiter observations.

Deconvolution

The final method of improving angular resolution is deconvolution. In its simplest form, atmospheric turbulence blurs the image of the object by convolving the true image with an effective point spread function (PSF). If we have a blurred image and we know the PSF, then theoretically we can simply deconvolve the measured image to recover the original image. Unfortunately, in practice things are not that simple since both the PSF and the blurred image include random noise, and with the simplest deconvolution methods this noise can propagate through to yield enormous errors in the deconvolved image. This phenomenon is known as ill-conditioning, and is also encountered when trying to retrieve vertical atmospheric profiles from measured IR spectra (as we shall see in Section 7.10). Practical deconvolution routines must somehow constrain the deconvolved solution to prevent noise error building up and a commonly used technique is the Richardson-Lucy (RL) deconvolution algorithm (e.g., Sromovsky et al., 2001a). If we represent the point spread function (PSF) as P{i | j) (where the PSF represents the fraction of light from true pixel j which gets scattered into pixel i) then the noiseless blurred image I(i) is formed from the

Figure 7.8. Two images of Uranus recorded with the Keck Observatory in AO mode, showing the additional improvement to resolution that can be achieved by using deconvolution. Courtesy of Keck Observatory, California Association for Research in Astronomy, and Larry Sromovsky.

Improving contrast with deconvolution:

Original AO image using H filter.

Deconvolved image.

unblurred image 0( j) as

The RL algorithm then takes the n th estimate of the unblurred image and improves it by the iteration equation

where D(i) is the observed image; and In(i) is the nth fit to the blurred image constructed using On(j) in Equation (7.11). Deconvolution algorithms may be used to improve the spatial resolution of images recorded by ground-based telescopes without adaptive optics, and may also be used to further process images recorded with adaptive optics, as is shown in Figure 7.8.

Interferometry

Interferometry is a technique routinely used in radio astronomy, and more recently microwave observations, which is currently under development at several optical/IR observatories also. Assuming that the effects of atmospheric turbulence are negated by the use of adaptive optics at visible wavelengths, or by observing at much longer wavelengths, the angular resolution of a single telescope is diffraction-limited by the diameter of the entrance aperture as

where D is the diameter of the entrance aperture; and A is the wavelength. The 1.22 factor comes from the fact that the entrance aperture is assumed to be circular and dd is specifically the angle between the center and first minimum of the Airy function. Hence, as we go to longer wavelengths, the aperture size required to achieve a specific angular resolution increases linearly with wavelength, which means that it is simply impractical to build a single-dish radio telescope with the same angular resolution as optical telescopes. However, an alternative approach is to use several telescopes, spaced over large distances, and combine their signals together with the appropriate phase delay to simulate, in effect, a giant mirror of diameter equal to the maximum separation of individual telescopes. The details of the recombination of the signals are complex, but as a simple example, two telescopes of diameter D placed a distance L apart, would have an effective angular resolution in the direction parallel to the line connecting the telescopes of A/L, and an angular resolution of A/D perpendicular to this direction. To achieve high resolution at all angles, interferometers usually have several telescopes arranged in a "T" or "Y" shape and the telescopes may usually be placed at a variety of separations in order to increase sensitivity. The imaging properties of such arrangements are complicated, but effectively such interferometers have an angular resolution of A/L, where L is the maximum baseline, and a field of view of A/D.

7.3.3 Brightness

We saw in Section 6.7 that the thermally emitted radiance (W m-2 sr _1 (cm-1)-1) of the giant planets decreases rapidly as we go out through the solar system due to decreasing atmospheric temperatures. What was not explicitly stated, however, is that reflected solar radiance also drops rapidly as 1/D2, where D is the distance of the planet from the Sun, due to these planets' greater and greater distance from the Sun. This decrease in reflected and thermally emitted radiance affects all remote-sensing observations, not just ground-based ones, and makes remote observation increasingly difficult as we go from Jupiter to Neptune. However, some ground-based observations of the giant planets, such as microwave observations or some thermal-IR spectroscopic observations, which have limited angular resolution, are unable to resolve the disks of these planets, especially Uranus and Neptune. Where this is the case there is a further factor decreasing the measured disk-averaged irradiance of the giant planets due to the rapidly decreasing projected solid angle of these planets as we go out through the solar system. At opposition, when the Earth is closest to the planet, the solid angle is given by

where D is the planet's distance to the Sun in AU; DAU is 1 AU (in kilometers); and R is the planetary radius (also in kilometers). Hence, the observed irradiance is given by the calculated disk-averaged radiance multiplied by the above solid angle and so drops even more rapidly as we go from Jupiter to Neptune due both to the increase in D, and to the decrease in R. For example, at visible wavelengths the opposition magnitudes of Jupiter, Saturn, Uranus, and Neptune are, respectively, -2.7, 0.67, 5.52, and 7.84, where the magnitude m (designed to formalize the observing convention of a 100-fold decrease in irradiance when going from magnitude-1 to

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