An extension of Newton's apsidal precession theorem

Newton's apsidal precession theorem in Proposition 45 of Book I of the 'Principia' has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ( N , e ), where N is the index for the centripetal force law, for varying eccentricity e . The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of   N = 2  which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives  ∂θ/∂ N   and  ∂θ/∂ h   [where h ( N , e ) is the angular momentum] are analytically expressed in the neighbourhood of   N = 2  (case of the inverse square law). The value of  ∂θ/∂ N   is computed numerically as well for  1 ≤ N < 3  . The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for   N = 2  , can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The   N = 1  case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high-eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.     

双基地激光测风雷达回波信号仿真研究

本文针对传统的体散射模型并未考虑大气不均匀性对信号传输的影响等问题,通过引入垂直非均匀的大气参数改进了模型,并利用其建立了体目标的双基地激光测风雷达方程,仿真了侧向散射回波信号,并与单基地雷达进行了对比分析。研究表明:水平方向上,双基地激光测风雷达的回波信号分布特征与单基地雷达差异较大,其回波信号等值线在近地面为卵形线,随着探测高度的增加,回波信号等值线逐渐变为以主、被动雷达为焦点的椭圆形,并最终趋近于圆形;垂直方向上,双基地激光测风雷达的回波信号随高度衰减剧烈,近地面的回波能量约为10^-10 J,4 km高度的回波能量约为10^-15J,在中低层大气(0~10 km),回波信号中气溶胶散射占比大,在高层大气(10 km以上),分子散射占比大。     

Neutral sodium atoms release from the surface of the Moon induced by meteoroid impacts

In this work, we calculate the vapour and neutral Na production rates on the Moon, as due to the impacts of meteoroids in the radius range of 10⁻⁸–0.15 m. We limit our calculations to this size range, since meteoroids with radius larger than 0.15 m have not been found to be important for the production of the exosphere in a time interval comparable with that of the observations.
We have considered a new dynamical model of the flux of meteoroids at the heliocentric distance of the Moon, regarding objects in the radius range of 10⁻²–0.15 m. Instead, the flux of smaller meteoroids (radius range 10⁻⁸–10⁻² m) has been calculated using the two distributions adopted by Cintala and Love & Brownlee.
The results of our model are that (i) the neutral Na production rate is  ∼3–4.9 × 10⁴ atoms cm⁻² s⁻¹  , slightly larger than the previous estimates  (∼2–3 × 10⁴ atoms cm⁻² s⁻¹)  , and (ii) only about 6 per cent of neutral Na is produced by the impacts of meteoroids in the size range 10⁻³–0.15 m, whereas about 94 per cent of the Na comes from the  10⁻⁵–10⁻³ m  size range.     

Solar system constraints on multifield theories of modified dynamics

Any viable theory of modified Newtonian dynamics (MOND) as modified gravity is likely to require fields in addition to the usual tensor field of General Relativity. For these theories, the MOND phenomenology emerges as an effective fifth force probably associated with a scalar field. Here, I consider the constraints imposed on such theories by Solar system phenomenology, primarily by the absence of significant deviations from inverse-square attraction in the inner Solar system as well as detectable local preferred frame effects. The current examples of multifield theories can be constructed to satisfy these constraints and such theories lead inevitably to an anomalous non-inverse-square force in the outer Solar system.     

Model of optical turbulence profile at Cerro Pachón

New measurements of optical turbulence profile at the Cerro Pachón observatory in Chile are analysed jointly with previously published data to model the variations of the intensity and thickness of the ground layer and free atmosphere under a variety of observing conditions. This work is motivated by the need to predict statistically the performance of ground-layer adaptice optics. We find that the ground-layer profile can be represented by a decaying exponent with a scale height of 20–40 m, increasing to 100 m under bad conditions. The zone from 6 to 500 m contributes typically about 61 per cent to the total integral, the latter causing a median seeing of 0.77 arcsec. Turbulence integrals in the ground layer and in free atmosphere vary independently of each other, in 50 per cent of cases they deviate by less than 1.8 times from their respective median values. The existence of periods with low turbulence in the free atmosphere and their importance for adaptive optics is stressed.     

Electron density diagnostic potential of the Si x ion and its application in Procyon

Theoretical electron density sensitive line ratios   R ₁– R ₆  of Si  x soft X-ray emission lines are presented. We found that these line ratios are sensitive to electron density n _e, and the ratio R ₁ is insensitive to electron temperature T _e. For reliable determination of the electron density of laboratory and astrophysical plasmas, atomic data, such as electron impact excitation rates, are very important. Our results reveal that the discrepancy of the line ratios from different atomic data calculated with the distorted wave (DW) approximation and the R-matrix method is up to 19 per cent at   n _e= 2 × 10⁸ cm⁻³  . We applied the theoretical intensity ratio R ₁ to the Low Energy Transmission Grating Spectrometer (LETGS) spectrum of the solar-like star Procyon. By comparing the observed value (1.29) with the theoretical calculation, the derived electron density n _e is  2.6 × 10⁸ cm⁻³  , which is consistent with that derived from  (C  v < 8.3 × 10⁸ cm⁻³)  . When the temperature structure of the Procyon corona is taken into account, the derived electron density increases from   n _e= 2.6 × 10⁸  to  2.8 × 10⁸ cm⁻³  .