Variable-mass system

Résumé

In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law of motion directly to such a system. Instead, the time dependence of the mass m can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system. The general equation of variable-mass motion is written as :\mathbf{F}{\mathrm{ext}} + \mathbf{v}{\mathrm{rel}}\frac{\mathrm{d} m}{\mathrm{d}t} = m {\mathrm{d} \mathbf v \over \mathrm{d}t} where Fext is the net external force on the body, vrel is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body, and v is the velocity of the body. In astrodynamics, which deals with the mechanics of rockets, the term vrel is often called the effective exhaust velocity and denoted ve. Derivation There are different derivations for the variable-mass sys

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https://en.wikipedia.org/wiki/Variable-mass_system

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Covariant formulation of relativistic mechanics

Miguel Alexandre Ribeiro Correia

Accretion disks surrounding compact objects, and other environmental factors, deviate satellites from geodetic motion. Unfortunately, setting up the equations of motion for such relativistic trajectories is not as simple as in Newtonian mechanics. The principle of general (or Lorentz) covariance and the mass-shell constraint make it difficult to parametrize physically adequate 4-forces. Here, we propose a solution to this old problem. We apply our framework to several conservative and dissipative forces. In particular, we propose covariant formulations for Hooke???s law and the constant force and compute the drag due to gravitational and hard-sphere collisions in dust, gas, and radiation media. We recover and covariantly extend known forces such as Epstein drag, Chandrasekhar???s dynamical friction, and Poynting-Robertson drag. Variable-mass effects are also considered, namely, Hoyle-Lyttleton accretion and the variable-mass rocket. We conclude with two applications: (1) The free-falling spring, where we find that Hooke???s law corrects the deviation equation by an effective anti???de Sitter tidal force and (2) black hole infall with drag. We numerically compute some trajectories on a Schwarzschild background supporting a dustlike accretion disk.

AMER PHYSICAL SOC2022

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L(X)We present weak lensing and X-ray analysis of 12 low-mass clusters from the Canada-France-Hawaii Telescope Lensing Survey and XMM-CFHTLS surveys. We combine these systems with high-mass systems from Canadian Cluster Comparison Project and low-mass systems from Cosmic Evolution Survey to obtain a sample of 70 systems, spanning over two orders of magnitude in mass. We measure core-excised L-X-T-X, M-L-X and M-T-X scaling relations and include corrections for observational biases. By providing fully bias-corrected relations, we give the current limitations for L-X and T-X as cluster mass proxies. We demonstrate that T-X benefits from a significantly lower intrinsic scatter at fixed mass than L-X. By studying the residuals of the bias-corrected relations, we show for the first time using weak lensing masses that galaxy groups seem more luminous and warmer for their mass than clusters. This implies a steepening of the M-L-X and M-T-X relations at low masses. We verify the inferred steepening using a different high-mass sample from the literature and show that variance between samples is the dominant effect leading to discrepant scaling relations. We divide our sample into subsamples of merging and relaxed systems, and find that mergers may have enhanced scatter in lensing measurements, most likely due to stronger triaxiality and more substructure. For the L-X-T-X relation, which is unaffected by lensing measurements, we find the opposite trend in scatter. We also explore the effects of X-ray cross-calibration and find that Chandra calibration leads to flatter L-X-T-X and M-T-X relations than XMM-Newton.

Oxford University Press2015

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