We address the expressions for the rates of the Keplerian orbital elements within a two-body problem perturbed by the tides in both partners. The formulae for these rates have appeared in the literature in various forms, at times with errors. We reconsider, from scratch, the derivation of these rates and arrive at the Lagrange-type equations which, in some details, differ from the corresponding equations obtained previously by Kaula (1964).
We also write down detailed expressions for $da/dt$, $de/dt$ and $di/dt$, to order $e^4$. They differ from Kaula's expressions which contain a redundant factor of $M/(M+M^{\prime}),$ with $M$ and $M^{\prime}$ being the masses of the primary and the secondary. As Kaula was interested in the Earth-Moon system, this redundant factor was close to unity and was unimportant in his developments. This factor, however, must be reinstated when Kaula's theory is applied to a binary composed of partners of comparable masses.
We have found that, while it is legitimate to simply sum the primary's and secondary's inputs in $da/dt$ or $de/dt$, this is not the case for $di/dt$. So our expression for $di/dt$ differs from that of Kaula in two regards. First, the contribution due to the dissipation in the secondary averages out when the apsidal precession is uniform. Second, we have obtained an additional term which emerges owing to the conservation of the angular momentum: a change in the inclination of the orbit causes a change of the primary's plane of equator.
Post-Keplerian perturbations of the hyperbolic motion in the field of a rotating massive object. Analysis in terms of osculating and nonosculating (contact) elements
Orbital migration and circularization of tidal debris by Alfvén-wave drag: circumstellar debris and pollution around white dwarfs
相当一部分白矮星（WD）表现出以$\sim10^3$ - $10^7$ kg s $^{-1}$的速率持续吸积难熔元素的迹象，其中，37个WD被检测到携带尘埃碎片盘。这样的同时发生不仅需要行星材料的肥沃储存，而且还需要高占空比的金属输送。一般认为，这种物质可能来自太阳系类似的主带小行星或柯伊伯带天体。在这里，我们认为WD污染物的原始祖先作为一个人口的残留高偏心率的星子，去挥发过程中的恒星巨星阶段。相当于太阳系的长周期彗星，它们被分散到WD附近，受到来自剩余行星、银河系潮汐、经过的分子云和附近恒星的扰动。当这些物体在潮汐扰动极限内冒险时，它们会经历缩小。我们定量地展示了材料强度和重力如何决定破碎条件和碎片尺寸。此后，碎片的半长轴需要衰减至少$\sim$ 6个数量级，然后它们的成分才能最终吸积到WD的表面上。我们研究了这些碎片在WD周围的轨道演化，并表明WD的磁场在其近星点通道期间引起阿尔文波阻力，并迅速使其轨道圆化。这一过程可能是负责所观察到的吸积率的重元素和周围WD碎片盘的产生。一个推测性的暗示是，巨行星可能在WD的祖先周围很常见，它们今天可能仍然与一些WD绑定。
