Central flashes during stellar occultations. Effects of diffraction, interferences, and stellar diameter

Abstract

Central flashes occur during stellar occultations by solar system objects. We catalog diffraction effects on the flash with point-like stars, monochromatic waves and spherical transparent atmosphere. Diffraction involves the Huygens principle, the Sommerfeld lemma and the stationary phase method, while finite stellar diameter cases involve Clausius' theorem. For point-like stars, the central flash shape is that of the classical Poisson spot, but with larger height. For tenuous atmospheres that cannot focus the stellar rays at shadow center, the flash is amplified by the factor (R_0/r_0)^2 compared to the Poisson spot, where R0 and r0 are the object and the shadow radii, respectively. For denser atmospheres that can focus the rays at shadow center, the flash peaks at 2[(pi*R/lambda_F})^2]*phi0, where R is the central flash layer radius, lambda_F is the Fresnel scale and phi0 is the flux that would be observed at shadow center without focusing. For isothermal atmospheres with scale height H, the height is 2(R*H)(pi/lambda_F)^2. Fringes surrounding the central flash are separated by lambda_P=lambda_F^2/R, related to the separation between the primary and secondary stellar images. For a projected stellar diameter D*>lambda_P, the flash is described by complete elliptic integrals, and has full width at half maximum of 1.14D* and peak value 8H/D*. For Earth-based occultations by Pluto and Triton observed in the visible with point-like stars, diffraction causes flashes with very large heights ~10e4-10e5, spread over a very small meter-sized region in the shadow plane. In practice, the flash is usually smoothed by the stellar diameter, but still reaches high values of ~50 and ~200 during Pluto and Triton occultations, respectively. Diffraction dominates when using millimeter wavelengths or longer. Effects of departure from sphericity, atmospheric waves and stellar limb darkening are discussed.

Figures (15)

• Figure 1: Principle of a stellar occultation by the atmosphere of an opaque spherical body with radius $R_0$, replaced here by a disk perpendicular to the figure. Upper panel: Tenuous atmosphere case, where the rays grazing the surface of the occulter at $R_0$ cannot converge towards the shadow center, creating a dark shadow of radius $r_0$ (thick black line in the observer's plane) with a Poisson spot at its center in $O'$. Lower panel: Dense atmosphere case, where there exists a layer with radius $R_{\rm CF}$ that focuses the rays towards the shadow center, creating a central flash in $O'$. In the gray zone of radius $r'_0$, an observer at P receives a ray coming from the primary (resp. secondary ) image plotted as the red (resp. blue) dashed line.
• Figure 2: Quantities used in the wave optics calculations. A star at infinity at the left of the figure along the $OO'$ direction, sends a plane wave through the occulter atmosphere, considered as a thin phase screen $XOY$. A source point S in this screen, with polar coordinates $(R,\theta)$, emits an elementary wave towards the observer in the shadow plane at P, with an inclination angle $\chi$ with respect to the normal of the screen. If the atmosphere is spherical, the point P can be placed without loss of generality along the $O'x$ axis, at distance $r=|x|$ from the shadow center. The phase of the elementary wave received at P depends on both the phase shift induced by the atmosphere at S and by the distance $l$ travelled by the wave between S and P.
• Figure 3: Diametric profile of the shadow cast by an opaque circular mask of radius $R_0=10$ km, illuminated by a monochromatic wave that provides a Fresnel scale of $\lambda_{\rm F}=1.2$ km. The profile has been obtained through the numerical integration of Eq. \ref{['eq_amplitude']}, where $\varphi_{\rm a} (R) \equiv 0$. The blue profile shows the shadow in the limit of geometrical optics, while the black profile accounts for diffraction effects. The edge of the shadow displays the Fresnel fringes with typical spacings of $\lambda_{\rm F}$ (Eq. \ref{['eq_Fresnel_scale']}), while the shadow center hosts the Poisson spot that peaks at the value of the flux far away outside the shadow, normalized here to unity. The upper red curve shows a shifted and expanded view of the Poisson spot. The fringes around the Poisson spot have a spacing of $\lambda_{\rm F}^2/R_0 = 0.144$ km (Eq. \ref{['eq_inter_fringe']}).
• Figure 4: Star (orange disk) located behind an object of radius $R_0$, as seen by an observer at P (Fig. \ref{['fig_geo_occ']}). The star appears as a disk with radius $r_*$ projected at the body distance, its center being at distance $r$ from the projected body center. Upper panel: Case $r > r_*$, the atmosphere produces two images, a primary (resp. secondary) image sketched as shaded the region delimited by the red (resp. blue) line. In a given direction defined by the angle $\theta$, the two points A and B along the stellar limb have as images the points A' and B', respectively. Lower panel: Case $r < r_*$, the two stellar images merge and form an Einstein-Chwolson ring. The apparent stellar radius $r_*$ and the atmospheric scale height $H$ have been greatly exaggerated for better visibility. In real cases, the stellar images are much more compressed.
• Figure 5: Central flash resulting from a finite stellar radius $r_*$ projected at the body distance, in the geometrical optics regime. The plot has been generated by used Eq. \ref{['eq_phi_r_gt_r*']} (blue part) and Eq. \ref{['eq_phi_r_lt_r*']} (red part). The functions $E$ and $G$ are given in Eqs. \ref{['eq_G']} and Table \ref{['tab_CF_formulae']}. The values along the horizontal axis have been normalized to the stellar radius $r_*$, while the flux along the vertical axis has been normalized to its peak value $4(R/r_*) \phi_\perp(0)$ (Eq. \ref{['eq_max_CF_stellar_diam']}). The two bullets show the values $x/r_*= \pm 1.14$ where the flash reaches half of its maximum value. The dotted line shows the diverging flash profile produced by a point-like star in the geometrical optics approximation (Eq. \ref{['eq_flash_geo_optics_iso']}).