Radius Of Circular Orbit Research Articles

Analytic Theory for High-Inclination Orbitsin the Restricted Three-Body Problem

DOI: 10.2514/1.41628 This paper explores the analytical solution properties surrounding a hypothetical orbit in an invariant plane perpendicular to the line joining the two primaries in the circular restricted three-body problem. Assuming motion can be maintained in the plane, Jacobi’s integral equation can be analytically integrated, yielding a closed-form expression for the period and path of the third body expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane assumption cannot be maintained naturally. However, the hypothetical orbit is shown to satisfy Jacobi’s integral equation and thetangentialmotionequation exactlyandtheothertwomotionequationsapproximatelyin bounded-averagedand banded sense. More important, the hypothetical solution can be used as the basis for an iterative analytical solution procedure for the three-dimensional trajectory where corrections are computable in closed form. In addition, the in-plane assumption can be strictly enforced with the application of a modulated thrust acceleration which is expressible in closed form. Presented methodology is primarily concentrated on halo-class orbits. Nomenclature a = radius of circular orbit C = Jacobi’s constant G = universal gravitational constant J = Jacobi’s function m1 = mass of first primary m2 = mass of second primary,m2 � m1 r = position vector of third mass relative to center of mass r1 = position vector of first primary relative to center of mass r2 = position vector of second primary relative to center of mass r12 = distance between two primaries � 1 = position vector of third mass relative to first primary � 2 = position vector of third mass relative to second primary ! = angular velocity vector of rotating fame

Light Echoes in Kerr Geometry: A Source of High‐Frequency QPOs from Random X‐Ray Bursts

We propose that high-frequency quasi-periodic oscillations (HFQPOs) can be produced from randomly formed X-ray bursts (flashes) by plasma interior to the ergosphere of a rapidly rotating black hole. We show by direct computation of their orbits that the photons comprising the observed X-ray light curves, if due to a multitude of such flashes, are significantly affected by the black hole's dragging of inertial frames; the photons of each such burst arrive to an observer at infinity in multiple (double or triple), distinct "bunches" separated by a roughly constant time lag of Δ tlag/M ≃ 14, regardless of the bursts' azimuthal position. We argue that every other such "bunch" represents photons that follow trajectories with an additional orbit around the black hole at the photon circular orbit radius (a photon "echo"). The presence of this constant lag in the response function of the system leads to a QPO feature in its power density spectra, even though the corresponding light curve consists of a totally stochastic signal. This effect is by and large due to the black hole spin and is shown to gradually diminish as the spin parameter a decreases or the radial position of the burst moves outside the static limit surface (ergosphere). Our calculations indicate that for a black hole with Kerr parameter of a/M = 0.99 and mass of M = 10 M☉, the QPO is expected at a frequency of νQPO ∼ 1.3–1.4 kHz. We discuss the plausibility and observational implications of our model results, as well as its limitations.

Orbital Evolution of an IMBH in the Galactic Nucleus with a Massive Central Black Hole

Resent observations and theoretical interpretations suggest that intermediate-mass black holes (IMBHs; 100-105 M☉) are formed in the centers of young and compact star clusters born close to the center of their parent galaxy. Such a star cluster would sink toward the center of the galaxy, and at the same time stars are stripped out of the cluster by the tidal field of the parent galaxy. We investigated the orbital evolution of the IMBH, after its parent cluster is completely disrupted by the tidal field of the parent galaxy, by means of large-scale N-body simulations. We constructed a model of the central region of our galaxy, with a supermassive black hole (SMBH; ≥106 M☉) and Bahcall-Wolf stellar cusp, and placed an IMBH in a circular orbit of radius 0.086 pc. The IMBH sinks toward the SMBH through dynamical friction, but dynamical friction becomes ineffective when the IMBH reaches the radius inside which the initial stellar mass is comparable to the IMBH mass. This is because the IMBH kicks out the stars. This behavior is essentially the same as the loss-cone depletion observed in simulations of massive SMBH binaries. After the evolution through dynamical friction stalled, the eccentricity of the orbit of the IMBH goes up, resulting in the strong reduction of the merging timescale through gravitational wave radiation. Our result indicates that the IMBHs formed close to the galactic center can merge with the central SMBH in short time. The number of merging events detectable with DECIGO or BBO is estimated to be around 50 yr-1. The event rate for LISA would be similar or less, depending on the growth mode of IMBHs.

Particle motion around magnetized black holes: Preston-Poisson space-time

We analyze the motion of massless and massive particles around black holes immersed in an asymptotically uniform magnetic field and surrounded by some mechanical structure, which provides the magnetic field. The space-time is described by the Preston-Poisson metric, which is the generalization of the well-known Ernst metric with a new parameter, tidal force, characterizing the surrounding structure. The Hamilton-Jacobi equations allow the separation of variables in the equatorial plane. The presence of a tidal force from the surroundings considerably changes the parameters of the test particle motion: it increases the radius of circular orbits of particles and increases the binding energy of massive particles going from a given circular orbit to the innermost stable orbit near the black hole. In addition, it increases the distance of the minimal approach, time delay, and bending angle for a ray of light propagating near the black hole.

Orbital dynamics of Sun-facing solar sails under the constraint of constant sail temperature

The orbital dynamics of Sun-facing solar sails is investigated considering a constraint of constant sail temperature at the limit of the sail material. Although solar sails can normally be articulated so as to provide thrust with both a transverse and radial component, a Sunfacing attitude with the center of solar pressure behind the center of gravity may be preferred for very large or gossamer sails in order to achieve Sun-facing attitude stability. The proposed Sun-facing solar sails are applicable to space weather and geo-storm warning missions for monitoring the inner solar system environment by in-situ measurement of solar wind plasma and high-energy particle events. Constraining the temperature of the sail to the temperature limit of the sail material allows the innermost circular orbits to be attained thereby maximizing scientific returns. The stability of the heliocentric circular orbit under such radial thrust with the constant temperature constraint is investigated, and the stability conditions are obtained as functions of the radius of circular orbit and the solar sail lightness number accounting for optical/thermal properties.

Circular orbits, ISCO, periastron shift, and gravitomagnetic clock effect in Kerr field: a study using effective one-body approach

Constraint Hamiltonian dynamics is used in an effective one-body approach to study relativistic gravitational two-body system with spin–orbit interaction. Radii of circular orbits, approximate location of innermost stable circular orbit (ISCO), periastron shift, and gravitomagnetic clock effect in the orbit of a non-spinning body around a slowly spinning body whose gravitational field is described by a limiting Kerr metric are derived.

On the tidal time delay of the Earth

We present an expression for the tidal time delay of the Earth, that takes into account paleonthological observations. A semi-empirical model of the dynamics of the Earth-Moon system is obtained considering that the energy dissipation and the time delay are not constant in time. In a previous work, we established a similar model by taking into account only the dynamics of a circular orbit radius for the Moon. In the present paper, three differential equations describe the dynamics of three variables, namely, the Moon's orbital mean radius, the eccentricity of the orbit, and the inclination of the Moon's orbital plane with respect to the inertial plane which is orthogonal to the total angular momentum of the Earth-Moon system. In this sense, the Earth-Moon system is considered isolated. We compare our picture with that one which considers the tidal time delay as constant in time and show that our approach is consistent with the modern theories concerning the Earth-Moon system formation.

Distant Companions and Planets around Millisecond Pulsars

We present a general method for determining the masses and orbital parameters of binary millisecond pulsars with long orbital periods (Porb 1 yr), using timing data in the form of pulse frequency derivatives. Our method can be used even when the available timing data cover only a small fraction of an orbit, but it requires high-precision measurements of up to five successive derivatives of the pulse frequency. With five derivatives a complete determination of the mass and orbital parameters is in principle possible (up to the usual inclination factor sin i). With less than five derivatives, only partial information can be obtained, but significant constraints can sometimes be placed on, e.g., the mass of the companion. We apply our method to analyze the properties of the second companion in the PSR B1620-26 triple system. We use the latest timing data for this system, including a recent detection of the fourth time derivative of the pulse frequency, to constrain the mass and orbital parameters of the second companion. We find that all possible solutions have a mass m2 in the range 2.4 × 10-4 M☉ ≤ m2 sin i2 ≤ 1.2 × 10-2 M☉, i.e., almost certainly excluding a second companion of stellar mass and suggesting instead that the system contains a planet or brown dwarf. To further constrain this system, we have used preliminary measurements of the secular perturbations of the inner binary. Using Monte Carlo realizations of the triple configuration in three dimensions, we find the most probable value of m2 to be 0.01 ± 0.005 M☉, corresponding to a distance of 38 ± 6 AU from the center of mass of the inner binary (the errors indicate 80% confidence intervals). We also apply our method to analyze the planetary system around PSR B1257+12, where a distant, giant planet may be present in addition to the three well-established Earth-mass planets. We find that the simplest interpretation of the frequency derivatives implies the presence of a fourth planet with a mass of ~100 M⊕ in a circular orbit of radius ~40 AU.

An eclipsing millisecond pulsar in the globular cluster Terzan 5

WE HAVE discovered an eclipsing binary millisecond pulsar in the globular cluster Terzan 5. This, the second known eclipsing binary pulsar after PSR1957 + 20, has a pulse period of 11.56 ms and a very short orbital period of 1.8 hours. In contrast to PSR1957 + 20, where the eclipses occupy about 10 per cent of the orbital period1, the eclipse duration in this pulsar is very variable and never less than one-third of the orbital period. The pulsar is in a circular orbit of radius 0.11 light seconds, which implies a minimum companion mass of 0.089 solar masses, about four times the companion mass of PSR1957 + 20. Timing observations suggest an identification of the pulsar with a variable continuum source locate about 30 arcsec west of the cluster centre. These observations and the variable eclipse duration show that the eclipse is due to absorption or scattering in a tenuous wind which flows from the companion star. We have also detected a second pulsar in the direction of Terzan 5. This pulsar, which has a period of 442 ms,may also be a cluster member, but is more likely to be a foreground object.

Laminar symmetry-plane boundary layer on a sharp spinning body at incidence.

Let the true values of the orbital elements be a = 1500 miles, e = 0.1, i = 1.4, Q = - 1.0, co = 0.6, tt = 0 sec. The y-axis is in the direction of the moon's motion with the x-axis completing a right-handed system. The moon is assumed to be in a circular orbit of radius 239,000 miles with an angular speed of 0.000002646659 rad/sec. The observing station is placed at the center of the Earth. \JL = 1175.525 miles3/sec2. These values give $! = -0.95247, s2 = 0.80792, s3 = 237611.9, s3 = 0.00058475. Substituting in the formulas we obtain approximate values for the orbital elements: a = 1500.86, e = 0.1027, co = 0.6448, sin/ = 0.98930 (true value is 0.98545). From the true values we find t = 4.133 when s is first equal to w. Thus tt = 4.133 sec, the true value being 0. Completing and Improving the Solution Having found approximate values for the five elements, we can choose a sign for cosz and guess a value for the sixth element Q. With six elements we can integrate the equations of motion for the lunar satellite with major perturbations included, using rt as the starting time. As the integration proceeds residuals are produced by comparing integrated values with observed values of range and range rate. Using standard statistical theory we form a function of the residuals and the orbital elements which is to be minimized by varying Q while holding the other five elements fixed at the values found by the method previously described. This minimization problem is not difficult since it is one-dimensional. We assume data available from a sufficient number of revolutions of the satellite to determine Q. The procedure described in this paragraph must be repeated with the sign of cos i changed. With approximations for six orbital elements we can compute approximate values of uz, uz, and uz at the times associated with a set of range and range rate measurements taken during a pass of the satellite in front of the moon. The procedure for finding five elements can now be repeated, using the following exact equations. z = suz — wz, z = suz + suz— vvz z = suz + 2sii_ + su. — w7

Ejection of Matter and Gravitational Radiation from Orbiting Bodies

IF a particle of mass m is in a circular orbit of radius r around a central body of mass M ≫ m and if M suddenly undergoes a decrease of mass, carried to infinity by massless particles, say, the orbit of m will be perturbed as the shell of emitted particles passes its radius. The question of the escape of m and of its exact subsequent motion has been discussed1,2 and in this communication we turn our attention to a related question.

Identification of Cen X-3

X-RAY observations of the pulsating X-ray source Cen X-3 indicate1 that it is a component of an eclipsing binary with a circular orbit of radius 0.08 AU. The position and period for Cen X-3 and for its suggested optical counterpart, LR Cen, are given in Table 1. There is a slight discrepancy in the agreement of the positions and periods. Observations are under way to redetermine the optical period.