An Improvement over Ray Optics: Airy’s Rainbow

This chapter discusses the optics of the rainbow as theorized by Sir George Biddle Airy. Airy developed a theory that quantified the dependence upon the raindrop size of the rainbow's angular width and angular radius as well as the spacing of the supernumeraries. The Airy theory also predicted a non-zero distribution of light intensity in Alexander's dark band and a finite intensity at the angle of minimum deviation. However, it could not predict precisely the angular position of many laboratory-generated rainbows, leading several mathematical physicists to seek a more complete theory of scattering. The chapter examines the Airy approximation by performing calculations, taking into account some ray prerequisites and the Airy wavefront. It also explains how colors are distributed in the Airy rainbow before concluding with a derivation for arbitrary p in the case of the Airy wavefront.

RX-J0852−4622: THE NEAREST HISTORICAL SUPERNOVA REMNANT – AGAIN

RX-J0852−4622, a supernova remnant, is demonstrated to be closer than 500 pc, based on the measurements of the angular radius, the angular expansion rate and the TeV g-ray flux. This is a new method of limiting the distance to any supernova remnant with hadronic induced TeV g-ray flux. The progenitor star of RX-J0852−4622 probably exploded in its blue supergiant wind, like SN 1987A, preceeded by a red supergiant phase. A cool dense shell, expected around the outskirts of the red wind, my have been identified. The distance (200 pc) and age (680 yr) of the supernova remnant, originally proposed, are supported.

Packing of twinned circles on a sphere

The twinned-circle problem is to pack 2 N non-overlapping equal circles forming N pairs of twins (rigidly connected neighbours) on a sphere so that the angular radius of the circles will be as large as possible. In the case that the contact graph(s) of the unconstrained circle packing support(s) at least one perfect matching, a complete solution to the twinned circles problem is found, with the same angular radius as the unconstrained problem. Solutions for N =2–12 pairs of twins are counted and classified by symmetry. For N =2–6 and 12, these are mathematically proven to be the best solutions; for N =7–11, they are based on the best known conjectured solutions of the unconstrained problem. Where the contact graph of the unconstrained problem has one or more rattling circles, the twinned problem is most easily solved by finding perfect matchings of an augmented graph in which each rattling circle is supposed to be simultaneously in contact with all its contactable neighbours. The underlying contact graphs for the unconstrained packings for N =2–12 are all Hamiltonian, guaranteeing the existence of perfect matchings, but Hamiltonicity is not a necessary condition: the first solution to the twins problem based on an example of a non-Hamiltonian contact graph occurs at N =16.

Masses of the central stars of Planetary Nebulae

The aim of this study is to derive the masses of the central stars (CSPN) for a large sample of the planetary nebulae (PN). These masses, M∗, are derived from the observed PN positions in three diagnostic diagrams and their comparison with evolutionary tracks of model PN. Two of the diagrams, namely LZan(H) versus TZan(H) and Mv versus Rneb, have already been used in numerous studies. The third one, SHβ versus SV, has recently been introduced by Górny, Stasińska & Tylenda (1996, hereinafter GST96). Here SHβ is the nebular surface brightness in Hβ and Sv is defined as Fv/(πθ2), where Fv is the stellar flux in the V band and θ is the observed nebular angular radius.

A Comparison of Nebular Distance Scales

Determination of the positions of central stars of planetary nebulae in the HR-diagram requires the knowledge of nebular distances. For almost all nebulae, these can only be given in terms of statistical scales. These scales have in common that they assume all nebulae to have the same structure (e.g. constant density) and that a unique ionized mass-radius relation exists. If the mass-radius relation is given by Mion = M0 · (R/R0)η, the distance d(pc) of planetary nebulae can be expressed as a function the de-reddened Hβ-flux (erg cm−2s−1) and the angular radius θ(arcsec): M0 and R0 are in solar masses and pc (Te = 10000 K, He/H = 0.1). The parameter η characterizes the distance scale: e.g. Shklovsky (1956) η = 0, Maciel L. Pottasch (1980) η = 1, Pottasch (1984) η = 3/2, Daub (1982) η = 5/3, and Kwok (1985) η = 9/4.

Covering a sphere by equal circles, and the rigidity of its graph

How must a sphere be covered by n equal circles so that the angular radius of the circles will be as small as possible? In this paper, conjectured solutions of this problem for n = 15 to 20 are given and some sporadic results for n > 20 (n = 22, 26, 38, 42, 50) are presented. The local optima are obtained by using a ‘cooling technique’ based on the theory of bar-and-joint structures. Thus the graph of the coverings by circles is considered as a spherical cable net in which the edge lengths are uniformly decreased, e.g. due to a uniform decrease in the temperature, until the graph becomes rigid and tensile stresses appear in the cables.

Random spherical triangles II: Shape densities

This paper gives the exact evaluation of the shape density on the shape space Σ(S2, 3) for a labelled random spherical triangle whose vertices are i.i.d.-uniform in a ‘cap' of S2 bounded by a ‘small' circle of angular radius ρ0.

Random spherical triangles II: Shape densities

This paper gives the exact evaluation of the shape density on the shape space Σ(S 2, 3) for a labelled random spherical triangle whose vertices are i.i.d.-uniform in a ‘cap' of S2 bounded by a ‘small' circle of angular radius ρ 0 .