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

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Volume 707 (March 2026)

A&A, 707 (2026) A208

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Table F.1

Parameters of the central flashes in the immediate vicinity of the shadow center.

Case with diffraction by an occulter with dense atmosphere, point-like star and monochromatic wave
	General case	Isothermal atmosphere, near centrality (r ≪ R_CF)
Radial profile, $ϕ_{D i f f} (r ≲ \frac{λ_{F}^{2}}{R_{C F}})$ $Mathematical equation: $\displaystyle \phi_{\rm Diff} \left(r \lesssim \frac{\lambda_{\rm F}^2}{R_{\rm CF}} \right)$$	$2 π^{2} {(\frac{R_{C F}}{λ_{F}})}^{2} J_{0}^{2} (\frac{π R_{C F} r}{λ_{F}^{2}}) ϕ_{⊥} (0)$ $Mathematical equation: $\displaystyle 2\pi^2 \left( \frac{R_{\rm CF}}{\lambda_{\rm F}} \right)^2 J_0^2 \left( \frac{\pi R_{\rm CF} r}{\lambda_{\rm F}^2} \right) \phi_\perp(0)$$	$2 π^{2} (\frac{R_{C F} H}{λ_{F}^{2}}) J_{0}^{2} (\frac{π R_{C F} r}{λ_{F}^{2}})$ $Mathematical equation: $\displaystyle 2\pi^2 \left( \frac{R_{\rm CF} H}{\lambda_{\rm F}^2} \right) J_0^2 \left( \frac{\pi R_{\rm CF} r}{\lambda_{\rm F}^2} \right)$$
Peak value, Φ_Diff(0)	$2 π^{2} {(\frac{R_{C F}}{λ_{F}})}^{2} ϕ_{⊥} (0)$ $Mathematical equation: $\displaystyle 2\pi^2 \left( \frac{R_{\rm CF}}{\lambda_{\rm F}} \right)^2 \phi_\perp(0)$$	$2 π^{2} (\frac{R_{C F} H}{λ_{F}^{2}})$ $Mathematical equation: $\displaystyle 2\pi^2 \left( \frac{R_{\rm CF} H}{\lambda_{\rm F}^2} \right)$$
Diameter of first dark fringe	$1.53 (\frac{λ_{F}^{2}}{R_{C F}})$ $Mathematical equation: $\displaystyle 1.53 \left( \frac{\lambda_{\rm F}^2}{R_{\rm CF}} \right)$$	$1.53 (\frac{λ_{F}^{2}}{R_{C F}})$ $Mathematical equation: $\displaystyle 1.53 \left( \frac{\lambda_{\rm F}^2}{R_{\rm CF}} \right)$$
Asymptotic form $ϕ_{D i f f} (r ≳ \frac{λ_{F}^{2}}{R_{C F}})$ $Mathematical equation: $\displaystyle \phi_{\rm Diff} \left(r \gtrsim \frac{\lambda_{\rm F}^2}{R_{\rm CF}} \right)$$	$ϕ_{1} + ϕ_{2} + 2 \sqrt{ϕ_{1} ϕ_{2}} \sin (φ_{2} - φ_{1})$ $Mathematical equation: $\displaystyle \phi_1 + \phi_2 + 2\sqrt{\phi_1 \phi_2} \sin \left( \varphi_2 - \varphi_1 \right)$$	$\frac{2 H}{r} + \frac{2 H}{r} \sin (\frac{2 π R_{C F} r}{λ_{F}^{2}})$ $Mathematical equation: $\displaystyle \frac{2H}{r} + \frac{2H}{r} \sin \left( \frac{2\pi R_{\rm CF} r}{\lambda_{\rm F}^2} \right)$$
Poisson fringe spacing, λ_P	$\frac{λ_{F}^{2}}{R_{C F}}$ $Mathematical equation: $\displaystyle \frac{\lambda_{\rm F}^2}{R_{\rm CF}}$$	$\frac{λ_{F}^{2}}{R_{C F}}$ $Mathematical equation: $\displaystyle \frac{\lambda_{\rm F}^2}{R_{\rm CF}}$$
Flash layer thickness, Δ R_Diff	$\sim λ_{F} \sqrt{ϕ_{⊥} (0)}$ $Mathematical equation: $\displaystyle \sim \lambda_{\rm F} \sqrt{\phi_\perp(0)}$$	$\sim λ_{F} \sqrt{\frac{H}{R_{C F}}}$ $Mathematical equation: $\displaystyle \sim \lambda_{\rm F} \sqrt{\frac{H}{R_{\rm CF}}}$$
Case with finite stellar diameter in the geometrical optics approximation and uniform stellar disk
Radial profile, Φ_Star(r ≥ r_*)	$(\frac{8 R}{π r_{}}) G (\frac{r_{}}{r}) ϕ_{⊥} (0)$ $Mathematical equation: $\displaystyle \left( \frac{8R}{\pi r_}\right) G\left(\frac{r_}{r}\right) \phi_\perp(0)$$	$(\frac{8 H}{π r_{}}) G (\frac{r_{}}{r})$ $Mathematical equation: $\displaystyle \left( \frac{8H}{\pi r_}\right) G\left(\frac{r_}{r}\right)$$
Radial profile, Φ_Star(r ≤ r_*)	$(\frac{8 R}{π r_{}}) E (\frac{r}{r_{}}) ϕ_{⊥} (0)$ $Mathematical equation: $\displaystyle \left( \frac{8R}{\pi r_}\right) E\left(\frac{r}{r_}\right) \phi_\perp(0)$$	$(\frac{8 H}{π r_{}}) E (\frac{r}{r_{}})$ $Mathematical equation: $\displaystyle \left( \frac{8H}{\pi r_}\right) E\left(\frac{r}{r_}\right)$$
Flash width, FWHM(Φ_Star)	1.14D_*	1.14D_*

Notes. See the definitions in Table 1. The effects due to limb darkening are described in Appendix E.

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