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Notes on the Barometric Formula
Published in Miguel A. Esteso, Ana Cristina Faria Ribeiro, A. K. Haghi, Chemistry and Chemical Engineering for Sustainable Development, 2020
For our atmosphere with T = 290 K, H = 8.5 km (the so-called scale height). Such a value overlaps with the value estimated by E. Halley1 by using the ratio of mercury to air densities, which is 10,800/1, and in this case we also have: H = 762 ×10,800 = 8.3 km.
Detectors
Published in C. R. Kitchin, Astrophysical Techniques, 2020
where R is the distance from the centre of the Earth, T the atmospheric temperature, M the mass of the Earth and m the mean particle mass for the atmosphere. Thus, the scale height increases with temperature, and so a given pressure will be reached at a greater altitude if the atmospheric temperature increases. The muons, however, are unstable particles and will have a longer time in which to decay if they are produced at greater heights. Thus, the muon and hence also the electron and positron intensity decreases as the atmospheric temperature increases. The relationship is given by () I(To)=I(T)e0.8(T−To)/T
Questions
Published in Michael de Podesta, Understanding the Properties of Matter, 2020
The quantity [kBT/mg] is known as the scale height, λ, of the atmosphere..The scale height is the height at which the pressure would fall to 1/e of its value at the ground. Evaluate this quantity for the Earth.
A minimal model for vertical shear instability in protoplanetary accretion disks
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
Ron Yellin-Bergovoy, Orkan M. Umurhan, Eyal Heifetz
An analytically tractable solution is obtained by assuming a neutrally-stratified exponential density profile in the form of , where is the mid-plane density and the density scale height, H, is assumed constant. Substituting this form for into equation (1d) we obtain where consequently, equation (2) has now an additional term in the right-hand side In order to properly account for stratification in the perturbation energy, we introduce the modified plane wave solution of the form of . Inserting into equation (22), we obtain the complex dispersion relation The equivalent VSI condition to equation (4) for the “exponential atmosphere” is derived in appendix B (see particularly (B.1), (B.8) and (B.9)), where , and . For infinite scale height H, equation (4) is recovered. For finite H however, the mean parcel's slope is smaller than the ratio k/m and by the same time the minimal critical slope for instability increases. Therefore, the anelastic effect tends to inhibit VSI.
Direct trajectory optimization framework for vertical takeoff and vertical landing reusable rockets: case study of two-stage rockets
Published in Engineering Optimization, 2019
Lin Ma, Kexin Wang, Zhijiang Shao, Zhengyu Song, Lorenz T. Biegler
Under these simplifying assumptions, the dynamic model is given as follows: where is the position vector, is the velocity vector, is the thrust vector, and is the drag force vector. is the gravitational parameter, is the gravity acceleration at sea level, and is the specific impulse of the engine. The drag force vector is defined as follows: where is the drag coefficient, is the reference area, and is the atmospheric density. The atmospheric density is defined as follows: where is the atmospheric density at sea level, is the altitude, is the equatorial radius of the Earth, and is the density scale height.
Augmented robust three-stage extended Kalman filter for Mars entry-phase autonomous navigation
Published in International Journal of Systems Science, 2018
Mengli Xiao, Yongbo Zhang, Zhihua Wang, Huimin Fu
During Mars entry, the vehicle is assumed to be a particle, and subject to gravity and aerodynamic forces. Moreover, if the Mars rotation and winds are neglected, a simplified 3-degree of freedom (3-DOF) dynamic could be defined with respect to the Mars-centred, Mars-fixed coordinate system (Levesque, 2006): where r denotes the distance from the centre of Mars to the centre of mass of the entry vehicle, θ and λ denote the longitude and the latitude, respectively, v is the velocity of the entry vehicle, γ is the flight-path-angle (FPA), ψ is the azimuth angle which is defined as a clockwise rotation angle starting at due north, φ is the bank angle which is defined as the angle about the velocity vector from the local vertical plane to the lift vector, and the unique control variable, D and L are the aerodynamic drag and lift accelerations, which are defined by where CD and CL are the aerodynamic drag and lift coefficients, respectively. CD and CL are the functions referred to the angle of attack, the sideslip angle and the Math, S is the reference surface area, m is the mass of the entry vehicle, is the lift-to-drag ratio, is called the ballistic coefficient, 0.5ρv2 denotes the dynamic pressure and ρ is the Mars atmospheric density which has been modelled experientially as follows: where ρ0 is the reference density, r0 = 3437.2km is the reference altitude and hs= 7.5km is the constant atmospheric scale height. Also, an inverse square gravitational acceleration is considered as where μ = GMMars is the Mars gravitational constant.