Pedal equation and Kepler kinematics

Kepler's laws is an appropriate topic which brings out the significance of pedal equation in Physics. There are several articles which obtain the Kepler's laws as a consequence of the conservation and gravitation laws. This can be shown more easily and ingeniously if one uses the pedal equation of an Ellipse. In fact the complete kinematics of a particle in a attractive central force field can be derived from one single pedal form. Though many articles use the pedal equation, only in few the classical procedure (without proof) for obtaining the pedal equation is mentioned. The reason being the classical derivations can sometimes be lengthier and also not simple. In this paper using elementary physics we derive the pedal equation for all conic sections in an unique, short and pedagogical way. Later from the dynamics of a particle in the attractive central force field we deduce the single pedal form, which elegantly describes all the possible trajectories. Also for the purpose of completion we derive the Kepler's laws.

Convective pattern formation in a thermally heated rotating annulus with magnetic central force field

<p>The earth, a sphere consisting of several layers like an onion is still up to now not fully understood. Gaining the fundamental knowledge to understand the mystery of global cell formation and large-scale convection in the interior or at the surface e.g. in our atmosphere is still of great interest from a meteorological point of view and of course in geophysics. However, laboratory experiments are still exposed to a significant problem – gravity. Establishing a radial force field e.g. in a sphere or annulus is still overpowered by gravity unless the experiment is carried out in a microgravity environment. Here, we show a potential application of a central force field induced by magnetic forces that acts on a magnetic fluid in a rotating thermally heated annulus to induce thermomagnetic convection and waves that are similar to the baroclinic annulus with the focus to study large scale atmospheric flow fields in a small laboratory system.</p><p> </p><p>Thermomagnetic convection is based on non-isothermal variation of fluid magnetisation induced e.g. by a temperature gradient in the presence of an external magnetic field. After Currie’s law colder magnetic fluid exhibits a larger fluid magnetisation and is therefore attracted to higher magnetic field intensities. This phenomenon is used to induced convection in a thermally heated annulus filled with a magnetisable ferro-magnetic fluid. Here, we study a 2-dimensional numerical problem geometry where the fluid is cooled at the inner and heated at the outer cylinder. The system is forced with an increasing central force field such that colder fluid is attracted towards the outer boundary when a critical threshold is exceeded – the critical magnetic Rayleigh number an equivalent non-dimensional parameter to the classical Rayleigh number for natural convection.</p><p> </p><p>Numerical results are obtained for two different radii ratios (0.35, 0.5). The parametric study included a range of magnetic Rayleigh numbers between 10<sup>3</sup> to 7.5x10<sup>5</sup> to induce a range of thermomagnetic convective cases. In addition, the thermally annulus is rotated at different speeds expressed via the Taylor number ranging from 10<sup>5</sup> to 10<sup>6</sup>. The observed flow fields reveal similar flow structures as seen in the classical baroclinic wave tank but have a different physically interpretation. The observed modes range from mode number 2 to 8 with stable symmetric to oscillatory and chaotic behaviours. The results are summarised in a regime diagram that is spanned in the thermally forcing and rotation speed space. This may be able to classify certain structures that are used to study atmospheric flow fields for different rotation and thermal forcing states e.g. planetary waves.</p>

Complementary numerical and experimental study in the baroclinic annulus for the microgravity experiment AtmoFlow

<p>The experimental investigation of large-scale flows on atmospheric circulation and climate such as Earth, Mars or even distant exoplanets are of great interest in geophysics. Gaining the fundamental knowledge of the origin of planetary waves or global cell formation is interesting from a meteorological point of view but up till now difficult to reproduce in laboratory scale. The limitation is based on the central force field which may be induced by the dielectrophoretic effect. However, the established radial force field is overpowered by the gravitational field unless experiments are conducted in a microgravity environment. The AtmoFlow project provides the possibility to study convective flow patterns in a spherical shell under microgravity conditions, planned after 2022, on the International Space Station (ISS) and is in fact the follow-up experiment of the GeoFlow project which served between 2008 and 2016 on the ISS.</p><p> </p><p>Without losing the overall focus of complex planetary atmospheres, the AtmoFlow experiment is able to model the intake and outtake of energy (e.g. radiation) and the rotational forcing via rotating or co-rotating boundaries. The gap is filled with a Fluor-based fluid with physical properties sensitive to temperature and electric fields. With an electric potential applied between the spherical shells a central force field is established that is based on the above mentioned dielectrophoretic effect. By adjusting rotation, thermal forcing and strength of the applied electric potential the AtmoFlow experiment can simulate different planetary atmospheres to investigate local pattern formation or global planetary cells. An interferometry system similar to the one used in the GeoFlow experiment uses the Wollaston shearing technique (WSI) to record the evolving temperature fields.</p><p> </p><p>To provide a benchmark solution for the experimentally recorded WSI interferograms a ground experiment is used to develop a validation method and to find the best postprocessing method for the AtmoFlow experiment. The ground experiment consists of a thermally forced baroclinic wave tank with a corresponding WSI setup and an infrared (IR) camera that are used to record the evolving temperature field. Here, we present first numerical simulations of the ground experiment that include the formation of the convective wave patterns and the numerical evaluated interferograms and IR pictures. The numerical calculated data will then be compared to the experimental recorded data to find a technique to best process the recorded WSI interferograms of the AtmoFlow project.</p>