As an extension of our previous paper, instead of the total deflection angle α, we will mainly focus on the discussion of measurable angle of the light ray ψP at the position of observer P in Kerr–de Sitter spacetime, which includes the cosmological constant Λ. We will investigate the contribution of the radial and transverse motion of the observer which are connected with radial velocity vr and transverse velocity bvϕ (b is the impact parameter) as well as the spin parameter a of the central object which induces the gravito-magnetic field or frame dragging and the cosmological constant Λ. The general relativistic aberration equation is employed to take into account the influence of motion of the observer on the measurable angle ψP. The measurable angle ψP derived in this paper can be applicable to the observer placed within the curved and finite-distance region in the spacetime. The equation of light trajectory will be obtained in such a sense that the background is de Sitter spacetime instead of Minkowski one. As an example, supposing the cosmological gravitational lensing effect, we assume that the lens object is the typical galaxy and the observer is in motion with respect to the lensing object at a recession velocity vr=bvϕ=vH=H0D (where H0 is a Hubble constant and D means the distance between the observer and the lens object). The static terms O(Λbm,Λba) are basically comparable with the second order deflection term O(m2), and they are almost one order smaller that the Kerr deflection −4ma/b2. The velocity-dependent terms O(Λbmvr,Λbavr) for radial motion and O(Λb2mvϕ,Λb2avϕ) for transverse motion are at most two orders of magnitude smaller than the second order deflection O(m2). We also find that even when the radial and transverse velocity have the same sign, asymptotic behavior as ϕ approaches 0 is different from each other, and each diverges to opposite infinity.

Assuming a static and spherically symmetric spacetime, we propose a novel concept of the total deflection angle of a light ray in terms of the optical geometry which is the Riemannian geometry experienced by the light ray. The total deflection angle is defined by the difference between the sum of internal angles of two triangles; one of the triangles lies on curved spacetime distorted by a gravitating body and the other on its background. The triangle required to define the total deflection angle can be realized by setting three laser-beam baselines as in planned space missions such as LATOR, ASTROD-GW, and LISA. Accordingly, the new total deflection angle is, in principle, measurable by gauging the internal angles of the triangles. The new definition of the total deflection angle can provide a geometrically and intuitively clear interpretation. Two formulas are proposed to calculate the total deflection angle on the basis of the Gauss-Bonnet theorem. It is shown that in the case of the Schwarzschild spacetime, the expression for the total deflection angle αSch reduces to Epstein-Shapiro's formula when the source of a light ray and the observer are located in an asymptotically flat region. Additionally, in the case of the Schwarzschild-de Sitter spacetime, the expression for the total deflection angle αSdS comprises the Schwarzschild-like parts and coupling terms of the central mass m and the cosmological constant Λ in the form of O(Λ m) instead of O(Λ/m). Furthermore, αSdS does not include the terms characterized only by the cosmological constant Λ.

Assuming a static and spherically symmetric spacetime, we propose a novel concept of the total deflection angle of a light ray. The concept is defined by the difference between the sum of internal angles of two triangles; one of the triangles lies on curved spacetime distorted by a gravitating body and the other on its background. The triangle required to define the total deflection angle can be realized by setting three laser-beam baselines as in planned space missions such as LATOR, ASTROD-GW, and LISA. Accordingly, the new total deflection angle is, in principle, measurable by gauging the internal angles of the triangles. The new definition of the total deflection angle can provide a geometrically and intuitively clear interpretation. Two formulas are proposed to calculate the total deflection angle on the basis of the Gauss--Bonnet theorem. It is shown that in the case of the Schwarzschild spacetime, the expression for the total deflection angle $\alpha_{\rm Sch}$ reduces to Epstein--Shapiro's formula when the source of a light ray and the observer are located in an asymptotically flat region. Additionally, in the case of the Schwarzschild--de Sitter spacetime, the expression for the total deflection angle $\alpha_{\rm SdS}$ comprises the Schwarzschild-like parts and coupling terms of the central mass $m$ and the cosmological constant $\Lambda$ in the form of ${\cal O}(\Lambda m)$ instead of ${\cal O}(\Lambda/m)$. Furthermore, $\alpha_{\rm SdS}$ does not include the terms characterized only by the cosmological constant $\Lambda$.

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild–de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle \(\alpha \) of the light ray by constructing a quadrilateral \(\varSigma ^4\) on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) determined by the optical metric \(\bar{g}_{ij}\). On the basis of the definition of the total deflection angle \(\alpha \) and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle \(\alpha \); (1) the angular formula that uses four angles determined on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) or the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\) being a slice of constant time t and (2) the integral formula on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) which is the areal integral of the Gaussian curvature K in the area of a quadrilateral \(\varSigma ^4\) and the line integral of the geodesic curvature \(\kappa _g\) along the curve \(C_{\varGamma }\). As the curve \(C_{\varGamma }\), we introduce the unperturbed reference line that is the null geodesic \(\varGamma \) on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting \(\varGamma \) vertically onto the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\). We demonstrate that the two formulas give the same total deflection angle \(\alpha \) for the Schwarzschild and the Schwarzschild–de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein–Shapiro’s formula when the source S and the receiver R of the light ray are located at infinity. In addition, in the Schwarzschild–de Sitter case, there appear order \({\mathscr {O}}(\varLambda m)\) terms in addition to the Schwarzschild-like part, while order \({\mathscr {O}}(\varLambda )\) terms disappear.

We revisit the role of the cosmological constant Λ in the deflection of light by means of the Schwarzschild–de Sitter/Kottler metric. In order to obtain the total deflection angle α, the time transfer function approach is adopted, instead of the commonly used approach of solving the geodesic equation of photon. We show that the cosmological constant does appear in expression of the deflection angle, and it diminishes light bending due to the mass of the central body M. However, in contrast to previous results, for instance, that by Rindler and Ishak (Phys. Rev. D. 2007), the leading order effect due to the cosmological constant does not couple with the mass of the central body M.

We will comment on the perihelion/periastron advance of celestial bodies due to the cosmological constant Λ. It is well known that the cosmological constant Λ causes the perihelion/periastron shift; however, there seems to still exist a discrepancy among the various derived precession formulae. We will point out that the expression \(\Delta\omega_{\varLambda} = (\pi c^{2} \varLambda a^{3}/(GM))\sqrt{1 - e^{2}}\) is the general formula for any orbital eccentricity e and the expression Δω Λ =(πc 2 Λa 3/(GM))(1−e 2)3 comes from the nearly circular (e≪1) approximation.

Context. The quasi-Hilda comets (QHCs), being in unstable 3:2 Jovian mean motion resonance, are considered a major cause of temporary satellite capture (TSC) by Jupiter. Though the QHCs may be escaped Hilda asteroids, their origin and nature have not yet been studied in suffi cient detail. Of particular interest are long TSCs/orbiters. Orbiters - in which at least one full revolution about the planet is completed - are rare astronomical events; only four have been known to occur in the last several decades. Every case has been associated with a QHC: 82P/Gehrels 3; 111P/Helin-Roman-Crockett; P/1996 R2 (Lagerkvist); and the possibly QHC-derived D/1993 F2 (Shoemaker-Levy 9, SL9). Aims. We focus on long TSC/orbiter events involving QHCs and Jupiter. Thus we survey the known QHCs, searching for further long TSCs/orbiters over the past century. Methods. First, we confirmed the long TSC/orbiter events of 82P, 111P, and 1996 R2 in order to test our method against previous work, applying a general N-body Newtonian code. We then used the same procedure to survey the remaining known QHCs and search for long TSC/orbiter events.

We will investigate the influence of the inhomogeneity of the Universe, especially that of the Lemaître–Tolman–Bondi (LTB) model, on a gravitationally bound local system such as the solar system. We concentrate on the dynamical perturbation to the planetary motion and derive the leading order effect generated from the LTB model. It will be shown that there appear not only a well-known cosmological effect arisen from the homogeneous and isotropic model, such as the Robertson–Walker (RW) model, but also the additional terms due to the radial inhomogeneity of the LTB model. We will also apply the obtained results to the problem of secular increase in the astronomical unit, reported by Krasinsky and Brumberg (2004), and imply that the inhomogeneity of the Universe cannot have a significant effect for explaining the observed dAU/dt = 15 ±4  [m/century].

We revisit the effect of cosmological constant Lambda on the light deflection and its role in the cosmological lens equation. First, we reexamine the motion of photon in the Schwarzschild spacetime, and explicitly describe the trajectory of photon and deflection angle alpha up to the second order in G. Then the discussion is extended to the contribution of the cosmological constant Lambda in the Schwarzschild-de Sitter or Kottler spacetime. Contrary to the previous arguments, we emphasize the following points: (a) the cosmological constant Lambda does appear in the orbital equation of light, (b) nevertheless the bending angle of light alpha does not change its form even if Lambda not equal 0 since the contribution of Lambda is thoroughly absorbed into the definition of the impact parameter, and (c) the effect of Lambda is completely involved in the angular diameter distance D-A.

We attempt to calculate the gravitational time delay in a time-dependent gravitational field, especially in McVittie spacetime, which can be considered as the spacetime around a gravitating body such as the Sun, embedded in the FLRW (Friedmann-Lema\^itre-Robertson-Walker) cosmological background metric. To this end, we adopt the time transfer function method proposed by Le Poncin-Lafitte {\it et al.} (Class. Quant. Grav. 21:4463, 2004) and Teyssandier and Le Poncin-Lafitte (Class. Quant. Grav. 25:145020, 2008), which is originally related to Synge's world function $\Omega(x_A, x_B)$ and enables to circumvent the integration of the null geodesic equation. We re-examine the global cosmological effect on light propagation in the solar system. The round-trip time of a light ray/signal is given by the functions of not only the spacial coordinates but also the emission time or reception time of light ray/signal, which characterize the time-dependency of solutions. We also apply the obtained results to the secular increase in the astronomical unit, reported by Krasinsky and Brumberg (Celest. Mech. Dyn. Astron. 90:267, 2004), and we show that the leading order terms of the time-dependent component due to cosmological expansion is 9 orders of magnitude smaller than the observed value of $d{\rm AU}/dt$, i.e., $15 \pm 4$ ~[m/century]. Therefore, it is not possible to explain the secular increase in the astronomical unit in terms of cosmological expansion.

We phenomenologically developed a propagation model of high energy galactic cosmic rays. We derived the analytical solutions by adopting the semi-empirical diffusion equation, proposed by Berezinskii et al. (1990) and the diffusion tensor proposed by Ptuskin et al. (1993). This model takes into account both the symmetric diffusion and the antisymmetric diffusion due to the particle Hall drift. Our solutions are an extension of the model developed by Ptuskin et al. to a two-dimensional two-layer (galactic disk and halo) model, and they coincide completely with the solution derived by Berezinskii et al. in the absence of antisymmetric diffusion due to Hall drift. We showed that this relatively simple toy model can be used to explain the variation in the exponent of the cosmic ray energy spectrum, γ, around the knee E ≈10 15 eV .

We investigated the influence of dark matter on light propagation in the solar system. We assumed the spherical symmetry of spacetime and derived the approximate solution of the Einstein equation, which consists of the gravitational attractions caused by the central celestial body, i.e. the Sun, and the dark matter surrounding it. We expressed the dark matter density in the solar system in the following simple power-law form, $\varrho(t, r) = \rho(t)(\ell/r)^k$, where $t$ is the coordinate time; $r$, the radius from the central body; $\ell$, the normalizing factor; $k$, the exponent characterizing $r$-dependence of dark matter density; and $\rho(t)$, the arbitrary function of time $t$. On the basis of the derived approximate solution, we focused on light propagation and obtained the additional corrections of the gravitational time delay and the relative frequency shift caused by the dark matter. As an application of our results, we considered the secular increase in the astronomical unit reported by Krasinsky and Brumberg (2004) and found that it was difficult to provide an explanation for the observed $d{\rm AU}/dt = 15 \pm 4 ~[{\rm m/century}]$.

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We investigated the light propagation by means of the Robertson-McVittie solution which is considered to be the spacetime around the gravitating body embedded in the FLRW (Friedmann-Lemaitre-Robertson-Walker) background metric. We concentrated on the time delay and derived the correction terms with respect to the Shapiro's formula. To relate with the actual observation and its reduction process, we also took account of the time transformations; coordinate time to proper one, and conversely. proper time to coordinate one. We applied these results to the problem of increase of astronomical unit reported by Krasinsky and Brumberg [Krasinsky. G.A., Brumberg, V.A., 2004. Celest. Mech. Dyn. Astrn. 90, 267]. However, we found the influence of the cosmological expansion on the light propagation does not give an explanation of observed value, dAU/dt = 15 +/- 4 [m/century] in the framework of Robertson-McVittie metric. (C) 2008 Elsevier B.V. All rights reserved.

Comet 147P/Kushida-Muramatsu, discovered in 1993, is shown by a numerical integration study to have been captured as a jovian satellite from 1949 to 1961. Completing two full revolutions around Jupiter and with a capture duration of 12 yr, it is ranked 3rd among known comets in both these respects. Coming through the region near the L2 libration point from the Centaur region beyond the jovian orbit, it escaped via L1 to the quasi-Hilda comet region, demonstrating the role of the latter region as a dynamical route to and from jovian temporary satellite capture, via the Hill’s sphere. Temporary Satellite Captures and Orbiters Only a few discovered comets have been known to be temporarily captured as satellites by Jupiter. Some, such as 39P/Oterma in 1936–38 [1], involve the comet flying through the region near Jupiter in a rather short time. In other cases, termed orbiters [2], at least one full revolution about Jupiter is completed and the capture phase may last about a decade or more. The known examples are 82P/Gehrels 3 [3], 111P/Helin-RomanCrockett [4], P/1996 R2 Lagerkvist [5], and most famously D/1993 F2 Shoemaker-Levy 9. Numerical integrations [6] suggested that the capture duration of D/1993 F2 lasted more than 50 yr which would be the longest of all these comets. Quasi-Hilda Comets and Kushida-Muramatsu Temporary satellite capture can be an intermediate state in the transfer of comets from outside to inside Jupiter’s orbit, inside to outside, or inside and back to inside [7]. Moreover, the inside region often corresponds dynamically to the Hilda asteroid zone. All the above orbiter examples are quasi-Hilda comets (QHCs), although this is just known statistically for D/1993 F2 since it was already orbiting Jupiter at the time of discovery and there are large uncertainties in tracing its exact orbital history. We verified the orbiter events of 82P, 111P and 1996 R2, and then integrated all the other objects in the QHC list [8] back 100 yr, a timescale over which the computed orbital evolution has a reasonable chance of being real. It is found [9] that Comet 147P/KushidaMuramatsu, discovered in 1993, orbited Jupiter as a satellite from 1949–61. The result is confirmed by integrating more than 200 orbital clones of 147P whose initial elements are [10] also consistent with observations.

We give an idea and the order-of-magnitude estimations to explain the recently reported secular increase of the Astronomical Unit (AU) by Krasinsky and Brumberg (2004). The idea proposed is analogous to the tidal acceleration in the Earth-Moon system, which is based on the conservation of the total angular momentum and we apply this scenario to the Sun-planets system. Assuming the existence of some tidal interactions that transfer the rotational angular momentum of the Sun and using reported value of the positive secular trend in the astronomical unit, $\frac{d}{dt}{AU} = 15 \pm 4 {(m/cy)}$, the suggested change in the period of rotation of the Sun is about $21 {ms/cy}$ in the case that the orbits of the eight planets have the same "expansion rate." This value is sufficiently small, and at present it seems there are no observational data which exclude this possibility. Effects of the change in the Sun's moment of inertia is also investigated. It is pointed out that the change in the moment of inertia due to the radiative mass loss by the Sun may be responsible for the secular increase of AU, if the orbital "expansion" is happening only in the inner planets system. Although the existence of some tidal interactions is assumed between the Sun and planets, concrete mechanisms of the angular momentum transfer are not discussed in this paper, which remain to be done as future investigations.

BER OF THE PGC? K. Ohtsuka, H. Arakida, T. Ito, M. Yoshikawa and D. J. Asher. Tokyo Meteor Network, 1-27-5 Daisawa, Setagaya-ku, Tokyo 1550032, JAPAN. E-mail: ohtsuka@jb3.so-net.ne.jp. Waseda University, Shinjuku-ku, Tokyo 169-8050, JAPAN. National Astronomical Observatory, Mitaka, Tokyo 181-8588, JAPAN. ISAS/JAXA, Sagamihara, Kanagawa 229-8510, JAPAN. Armagh Observatory, College Hill, Armagh, BT61 9DG, UK.