Variational and symplectic integrators for satellite relative orbit propagation including drag

Orbit propagation algorithms for satellite relative motion relying on Runge–Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior and degraded accuracy. Thus, attempts have been made to apply symplectic methods to integrate satellite relative motion. However, so far all these symplectic propagation schemes have not taken into account the effect of atmospheric drag. In this paper, drag-generalized symplectic and variational algorithms for satellite relative orbit propagation are developed in different reference frames, and numerical simulations with and without the effect of atmospheric drag are presented. It is also shown that high-order versions of the newly-developed variational and symplectic propagators are more accurate and are significantly faster than Runge–Kutta-based integrators, even in the presence of atmospheric drag.     

On the energy integral for first post-Newtonian approximation

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials \(W^j (j=1,2,3)\), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.     

Capture into first-order resonances and long-term stability of pairs of equal-mass planets

Massive planets form within the lifetime of protoplanetary disks, and therefore, they are subject to orbital migration due to planet–disk interactions. When the first planet reaches the inner edge of the disk, its migration stops and consequently the second planet ends up locked in resonance with the first one. We detail how the resonant trapping works comparing semi-analytical formulae and numerical simulations. We restrict to the case of two equal-mass coplanar planets trapped in first-order resonances, but the method can be easily generalized. We first describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using series expansions up to different orders in eccentricity as well as a non-expanded Hamiltonian. Then we show that during convergent migration the planets evolve along these families of equilibrium points. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict precisely analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that in most cases the resonant configurations achieved by migration become unstable after the removal of the protoplanetary disk. Here we probe the stability of the resonances as a function of planetary mass. For this purpose, we fictitiously increase the masses of resonant planets, adiabatically maintaining the low-amplitude libration regime until instability occurs. We discuss two hypotheses for the instability, that of a low-order secondary resonance of the libration frequency with a fast synodic frequency of the system, and that of minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter nevertheless play the decisive role in the destabilization of resonant pairs. We show evidence that as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.     

Development of a Lunar Scintillometer as part of the national large optical telescope site survey

Ground layer turbulence is a very important site characterization parameter used to assess the quality of an astronomical site. The Lunar Scintillometer is a simple and effective site-testing device for measuring the ground layer turbulence. It consists of a linear array of photodiodes which are sensitive to the slight variations in the moon’s brightness due to scintillation by the lower layers of the Earth’s atmosphere. The covariance of intensity values between the non-redundant photodiode baselines can be used to measure the turbulence profile from the ground up to a height determined by the furthest pair of detectors. The six channel lunar scintillometer that has been developed at the Indian Institute of Astrophysics is based closely on an instrument built by the team led by Andrei Tokovinin of Cerro Tololo Inter-American Observatory (CTIO), Chile (Tokovinin et al., Mon. Not. R. Astron. Soc. 404(3), 1186–1196 2010). We have fabricated the instrument based on the existing electronic design, and have worked on the noise analysis, an EMI (Electromagnetic Induction) resistant PCB design and the software pipeline for analyzing the data from the same. The results from the instrument’s multi-year campaign at Mount Saraswati, Hanle is also presented.     

The First Solar Seeing Profile Measurement with Two Apertures and Multiple Guide Regions

The sky brightness is a critical parameter for estimating the coronal observation conditions for a solar observatory. As part of a site-survey project in Western China, we measured the sky brightness continuously at the Lijiang Observatory in Yunnan province in 2011. A sky brightness monitor (SBM) was adopted to measure the sky brightness in a region extending from 4.5 to 7.0 apparent solar radii based on the experience of the Daniel K. Inouye Solar Telescope (DKIST) site survey. Every month, the data were collected manually for at least one week. We collected statistics of the sky brightness at four bandpasses located at 450, 530, 890, and 940 nm. The results indicate that aerosol scattering is of great importance for the diurnal variation of the sky brightness. For most of the year, the sky brightness remains under 20 millionths per airmass before local Noon. On average, the sky brightness is less than 20 millionths, which accounts for 40.41% of the total observing time on a clear day. The best observation time is from 9:00 to 13:00 (Beijing time). The Lijiang Observatory is therefore suitable for coronagraphs investigating the structures and dynamics of the corona.     

Fraile Detection of Solar $$-Modes by Fossat et al.

The internal gravity modes of the Sun are notoriously difficult to detect, and the claimed detection of gravity modes presented by Fossat et al. (Astron. Astrophys.604, A40, 2017) is thus very exciting. Given the importance of these modes for understanding solar structure and dynamics, the results must be robust. While Fossat et al. described their method and parameter choices in detail, the sensitivity of their results to several parameters was not presented. Therefore, we test the sensitivity of the results to a selection of the parameters. The most concerning result is that the detection vanishes when we adjust the start time of the 16.5-year velocity time-series by a few hours. We conclude that this reported detection of gravity modes is extremely fragile and should be treated with utmost caution.     

A Major Geoeffective CME from NOAA 12371: Initiation,CME–CME Interactions,and Interplanetary Consequences

In this article, we present a multi-wavelength and multi-instrument investigation of a halo coronal mass ejection (CME) from active region NOAA 12371 on 21 June 2015 that led to a major geomagnetic storm of minimum \(\mathrm{Dst} = -204\) nT. The observations from the Atmospheric Imaging Assembly onboard the Solar Dynamics Observatory in the hot EUV channel of 94 Å confirm the CME to be associated with a coronal sigmoid that displayed an intense emission (\(T \sim6\) MK) from its core before the onset of the eruption. Multi-wavelength observations of the source active region suggest tether-cutting reconnection to be the primary triggering mechanism of the flux rope eruption. Interestingly, the flux rope eruption exhibited a two-phase evolution during which the “standard” large-scale flare reconnection process originated two composite M-class flares. The eruption of the flux rope is followed by the coronagraphic observation of a fast, halo CME with linear projected speed of 1366 km?s^?1. The dynamic radio spectrum in the decameter-hectometer frequency range reveals multiple continuum-like enhancements in type II radio emission which imply the interaction of the CME with other preceding slow speed CMEs in the corona within \(\approx10\)?–?\(90~\mbox{R} _{\odot}\). The scenario of CME–CME interaction in the corona and interplanetary medium is further confirmed by the height–time plots of the CMEs occurring during 19?–?21 June. In situ measurements of solar wind magnetic field and plasma parameters at 1 AU exhibit two distinct magnetic clouds, separated by a magnetic hole. Synthesis of near-Sun observations, interplanetary radio emissions, and in situ measurements at 1 AU reveal complex processes of CME–CME interactions right from the source active region to the corona and interplanetary medium that have played a crucial role towards the large enhancement of the geoeffectiveness of the halo CME on 21 June 2015.     

The Problem of the Height Dependence of Magnetic Fields in Sunspots

To understand the physics of sunspots, it is important to know the properties of their magnetic field, and especially its height stratification plays a substantial role. There are mainly two methods to assess this stratification, but they yield different magnetic gradients in the photospheric layers. Determinations based on the several spectral lines of different formation heights and the slope of their profiles result in gradients of ?2 to ?3 G?km^?1, or even steeper. This is similar for the total magnetic field strength and for the vertical component of the magnetic field. The other option is to determine the horizontal partial derivatives of the magnetic field, and with the condition \(\operatorname{div} {{\boldsymbol {B}}} = 0\) also the vertical derivative is known. With this method, gradients of ?0.5 G?km^?1 and even shallower are obtained. Obviously, these results do not agree. If chromospheric spectral lines are included, only shallow gradients around ?0.5 G?km^?1 are obtained. Shallow gradients are also found from gyro-resonance measurements in the radio wave range 300?–?2000 GHz.Some indirect methods are also considered, but they cannot clarify the total picture. An analysis of a numerical simulation of a sunspot indicates a shallow gradient over a wide height range, but with slightly steeper gradients in deep layers.Several ideas to explain the discrepancy are also discussed. With no doubts cast on Maxwell’s equations, the first one is to look at the uncertainties of the formation heights of spectral lines, but a wider range of these heights would require an extension of the solar photosphere that is incompatible with observations and the theory of stellar atmospheres. Submerging and rising magnetic flux might play a role in the outer penumbra, if the resolution is too low to separate them, but it is not likely that this effect acts also in the umbra. A quick investigation assuming a spatial small scale structure of sunspots together with twist and writhe of individual flux tubes shows a reduction of the measured magnetic field strength for spectral lines sensitive to a larger height range. However, sophisticated investigations are required to prove that the explanation for the discrepancy lies here, and the problem of the height gradient of the magnetic field in sunspots is still not solved.     

Mid-term periodicities and heliospheric modulation of coronal index and solar flare index during solar cycles 22–23

We applied fast Fourier transform techniques and Morlet wavelet transform on the time series data of coronal index, solar flare index, and galactic cosmic ray, for the period 1986–2008, in order to investigate the long- and mid-term periodicities including the Rieger (\({\sim }130\) to \({\sim }190\) days), quasi-period (\({\sim }200\) to \({\sim }374\) days), and quasi-biennial periodicities (\({\sim }1.20\) to \({\sim }3.27\) years) during the combined solar cycles 22–23. We emphasize the fact that a lesser number of periodicities are found in the range of low frequencies, while the higher frequencies show a greater number of periodicities. The rotation rates at the base of convection zone have periods for coronal index of \({\sim }1.43\) years and for solar flare index of \({\sim }1.41\) year, and galactic cosmic ray, \({\sim }1.35\) year, during combined solar cycles 22–23. In relation to these two solar parameters (coronal index and solar flare index), for the solar cycles 22–23, we found that galactic cosmic ray modulation at mid cut-off rigidity (\(\hbox {Rc} = 2.43\hbox {GV}\)) is anti-correlated with time-lag of few months.