condition for stable circular orbit

Conditions of Acceptance of a Particle into the Acceleration Cycle. The proper use of equation 1 requires that θ = π. Questions/tasks 1. Angular momentum is quantized. Useful constant: Newton's gravitational constant is. 3. No circular 58 orbit for a free stone exists for r<3M. v orbit = 2 π r / T. v orbit = 2 π r / T. We substitute this into Equation 13.7 and rearrange to get. a stable circular orbit of radius r if rBo= r°2B= = Bzr dr, (4) the well-known bbtatron condition. ... the condition for stability of circular orbits will change considerably relative to commutative case. T = 2 π r 3 G M E. T = 2 π r 3 G M E. (b) For this special case, find the orbital equation r(Φ), where r and Φ are polar coordinates. (c) Now consider a … Thus, the stability criterion for a circular orbit of radius in a central force-field characterized by a radial force (per unit mass) function is. A 19-year-old BA final year student was killed, and her friend was injured, when the two-wheeler they were riding on met with an accident at Sector 35-43 dividing road on Thursday. 1.3. If ω2 > 0, the circular orbit is stable and the perturbation oscillates harmonically. of the accretion disk around a neutron star @1#. One complete repetition of the motion is called a cycle. The schedule called for first launch in 1972. A spacecraft in a circular orbit wishes to transfer to another circular orbit of quarter the radius by means of a tangential thrust to move into an elliptical orbit and a second tangential thrust at the opposite end of the ellipse to move into the desired circular orbit. Evolving quasiequilibrium (QE) binaries that begin at different separations, we bracket the location of the ISCO by distinguishing stable circular orbits from unstable plunges. The DCS393 Circular Saw with 6-1/2 in. (311) For example, consider an attractive power-law force function of the form. “Torsional strain” refers to the circular force along both the horizontal and vertical axes of the object under test. Compressive Strain. As an example, assuming an attractive potential with a power law U = k r n, the last equation gives us that is has stable orbits only for n > − 2. The Lagrange solution is stable only if one of the three masses is much greater than the other two. In the (b)Find the frequency of small oscillations ! In a two-body problem, I think the effective potential for a given angular momentum would be an annular trough, the bottom of which would indicate the the radius of the stable circular orbit (and the trough shape indicating stability). Everything else is provided by objects. Velocity is a vector, so in line 17 use this value for the y-component of the star’s velocity. able to maintain a stable circular orbit which does not enter the event horizon of a black hole; this is called an innermost stable circular orbit (ISCO) [14]. Assuming that the potential is attractive and a … From the equation of motion of the dust ... One necessary condition for quasi-stable exterior resonance traps is derived. Satellites, which are objects orbiting other celestial objects, are often found in a stable orbit motion where their velocity and trajectory form a consistent pattern of revolutions. Possible potentials responsible for stable circular ... We first set up the relativistic orbit equation for a particle in the central potential presumed to be dependent on the radial coordinate only. The condition for stable circular orbits is then V̂ = 0 and ∂∂rV̂ = 0, that is the function V̂ (r; Ẽ, L̃) has a double root at r = rc . In Section 3, we derive 3. The effective potential for both time-like and null cases have been studied. The electron rotates around the nucleus in a stable circular orbit, with uniform circular motion. Consider a satellite revolving around the earth in a circular orbit. Gravity provides the inward pull that keeps the satellite in orbit. The simplest kind of closed orbit is a circular one. Sketch the effective potential energy for the cases that n = 2, -1, and -3. They are connected by three identical springs of sti ness k 1 = k 2 = k 3 = k, as shown. This section treats only the idealized, uniform circular orbit of a planet such as Earth about a central body such as the Sun. This expression looks like the total energy of a particle moving in one dimension in a potential U eff (r). A collision of particles in the inner stable circular orbit is considered and the conditions for the escape of colliding particles from the vicinity of black hole are given. 2 π r. 2 π r in one period T. Using the definition of speed, we have. If it did, it would gain speed as it falls and zoom back out of it. Sketch them for the cases that n= 2, n= 1, and n= 7. orbital radius. In that paper a search was made for the inner most stable circular orbit in the absence of radiation reaction terms in the equations of motion. (b) Find the radius at which the particle (with given angular momentum l) can orbit at fixed radius. If ω2 < 0, the circular orbit is unstable and the perturbation grows exponentially. of the desired circular orbit? Describe conditions necessary for a satellite to remain in a stable circular orbit around Earth. For the geometric shape of the perturbed orbit, we write r = r 0 + η, and from (9.18) we 5 The orbit of \( E_2 \) is also stable; there is a minimum and maximum value of \( r \), which the comet will move between in some way. So we have r … ... Find the condition for which it moves with constant radial speed. Although the criterion for circular orbits is most elegantly expressed in terms of the e↵ective potential, sometimes it’s necessary to go back to our original potential V(r). II, we use the formalism of the effective potential to derive the conditions for the existence of circular orbits on the equatorial plane of the Kerr spacetime.Sec. We have proved in (Eq. 2.) We derive the conditions for the existence of the innermost stable circular orbit, marginally bound circular orbit and circular photon orbit in the background of Kerr–Newman–Taub–NUT (KNTN) spacetime. Assuming a circular orbit, the gravitational force must equal the centripetal force. So we have r … –3. In addition, we calculated the radius of the photon sphere and the angular radius of the shadow of the 4D EGB black hole. From the given equation of the orbit, the perigee occurs at è=0. (c) For the stable case, show that the period of small oscillations about the circular orbit is ˝ osc = ˝ orb= p n+ 2. An orbit with an eccentricity of zero is a circular orbit, while an orbit with an eccentricity of one would be highly elongated. f e f f ′ ( r) < 0, for orbit to be stable. $\begingroup$ With all of those conditions, pretty darn stable. the circular orbits are neutrally stable. 5. (b) Explain why the condition dV eff(r)/dr = 0 locates a circular orbit and show that for the general n case, when the system has angular momentum , the circular orbit radius is: € r o = (2/nµk) 1/(n+2). The adiabatic orbit condition (2) for a relativistic electron becomes e2 r2 = γm0ar = γm0 v2 θ r ≈ γm0 v2 r. (19) This can be thought of as the transform of the rest-frame relation eE r = dP r /dt upon noting that E r = γEr since the electric field is tranverse to the velocity, dt∗ = dt/γ, and dP r = dPr = γm0dvr. About a day later, the three-man crew would ride aboard a Saturn IB into orbit to link up with the Workshop-ATM cluster, thus beginning the manned portion of the mission. To find the period of a circular orbit, we note that the satellite travels the circumference of the orbit. It is shown that the orbit of the spinning particle is related to the spin of the test particle. The Innermost stable circular orbit (often called the ISCO) is the smallest circular orbit in which a test particle can stably orbit a massive object in general relativity. The location of the ISCO, the ISCO-radius ( r i s c o {\displaystyle r_{isco}} ), depends on the angular momentum (spin) of the central object. Calculate the velocity of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km? that viscous torques cannot act across the black hole horizon, and the location of the cusp is directly connected to the efficiency of accretion. Except for the thermodynamically stable 1T′-MoTe 2 and 1T′-WTe 2 (ref. The nature of the radiation force and solar wind drag is briefly summarized in Section 2. In the test-mass limit, well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes. The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. The last circular orbit is represented by a point. Like the orbital integral, the orbital differential equation describes the rela-tion between the radial and angular coordinates of an orbit, a relation from mechanics - mechanics - Circular orbits: The detailed behaviour of real orbits is the concern of celestial mechanics (see the article celestial mechanics). We determine the innermost stable circular orbit (ISCO) of binary neutron stars (BNSs) by performing dynamical simulations in full general relativity. The variable can be modified from code (more on this later). Period and Frequency. For the geometric shape of the perturbed orbit, we write r = r0 +η, and from (9.18) we obtain d2η dφ2 = µr4 0 ℓ2 F′(r 0) −3 η = −β2 η , (9.26) with β2 = 3+ dlnF(r) dlnr r0. 59 Comment 1. The nature of boundedness of orbits of a particle moving in a central force field is investigated. v orbit = 2 π r / T. v orbit = 2 π r / T. We substitute this into Equation 13.7 and rearrange to get. 4.Show that the equilibrium is stable under small displacements. Deduce conditions on the form of the magnetic field such that motion of (relativistic) electrons transverse to the central orbit is stable. Stability of Circular Orbits. Calculate the velocity of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km? of the accretion disk around a neutron star @1#. It looks like effective potential drops off at a distance as centrifugal force takes over. orbits. Angular momentum: L = m v r is conserved. Although the criterion for circular orbits is most elegantly expressed in terms of the e↵ective potential, sometimes it’s necessary to go back to our original potential V(r). in circular orbit about a common center of mass (circular restricted three-body problem). Assuming “no torque inner boundary conditions”, i.e. At least four infants died late Monday in a fire that broke out in the children’s ward of the Kamla Nehru Hospital, which is located within the premises of Bhopal’s state-run Hamidia Hospital. This paper is organized as follows. A circular orbit was thought to be stable when the outward centrifugal fbrce characterized by radius r. and speed u (rr.u2/r) on the electron perf-ectlir counterbalanced … Montgomery (2001) describes several other solutions that exist when all the particles have the ... with a circular orbit (adapted from Frank, King & Raine 2002). The innermost stable circular orbit (often called the ISCO) is the smallest marginally stable circular orbit in which a test particle can stably orbit a massive object in general relativity. A moon orbits a planet in a nearly circular orbit of radius R, as shown in the figure. This is very weak compared to the statement of Bertrand's Theorem. It is shown that the orbit of the spinning particle is related to the spin of the test particle. \( E_1 \) corresponds to a stable, circular orbit, as in the spring example. This is the condition for a circular orbit. The trajectories of … The evolution of magnetic fields frozen to a perfectly conducting plasma fluid around a Kerr black hole is investigated. ID:CM-U-147 Three particles of the same mass m 1 = m 2 = m 3 = mare constrained to move in a common circular path. As the satellite moves to an orbit of greater radius, PE becomes less negative, hence it increases. One of the most strong rea-sons is that in many sources, the maximum frequency of the kHz QPOs is in narrow range between 1.1 and 1.2 kHz, although they are thought to have very different mass accre-tion rates and magnetic fields. min:0: rad: 5: Latitude/X: Center point latitude (if no MAV_FRAME specified) / X coordinate according to MAV_FRAME. We nd that a free stone can 56 move (a) in a stable circular orbit only at r-coordinates greater than r= 6M, 57 or (b) in an unstable circular orbit from r= 6Mdown to r= 3M. Argue that if p n+ 2 is a rational number, these orbits are closed. for a three-quarter orbit set 270*Pi/180). Electrons describe circular paths. If the orbit is circular then , constant, and . Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Find the condition for a stable circular orbit to exist. This is a custom text field that can be filled with a variable name. With this extension, we calculated the radius of the innermost stable circular orbit (ISCO) for the timelike particle and found that this radius is a deceasing function of the GB coupling constant. with simple harmonic motion. 3.) < 2, ? Show that these orbits are stable to shifts along the z axis if. Obtain two constants of motion. The usual physics terminology for motion that repeats itself over and over is periodic motion, and the time required for one repetition is called the period, often expressed as the letter T. (The symbol P is not used because of the possible confusion with momentum. ) the innermost stable circular orbit ~ISCO! (312) where . Question: Consider a central force ()=−2− /. When the satellite is orbiting around the earth it possesses two types of mechanical energies. The DCD771 Drill/Driver has a high-speed transmission with 2-speeds (0 RPM to 450 RPM and 1,500 RPM) for a range of fastening and drilling applications. Line 16 is the magic. The radius rc together with the parameters Ẽ(rc ), L̃(rc ) corresponds to a stable circular orbit. Earth’s spin axis is tilted relative to the plane of its orbit, and the seasons are. The Orbit Yard Enforcer (TM) Motion-Activated Sprinkler protects landscapes and gardens from animals and pests. For circular motion, the acceleration will always have a non-positive radial component (a r) due to the change in direction of velocity, (it may be zero at the instant the velocity is zero). If ω2 < 0, the circular orbit is unstable and the perturbation grows exponentially. with simple harmonic motion. Do your sketches con rm this conclusion? In the accretion disk the-ory [15], the ISCO is regarded as one of the rotating black hole’s important features like the event horizon, ergosphere, Such a planet could exist for a few years, but its orbit would not be stable. Xfade Time is the time to cross-fade between this state and the next. In each situation, analyze the conditions required for a stable circular orbit. Circular orbits will be stable if they correspond to a minimum of the potential, and unstable if they correspond to a maximum. We show that the stationary state of the magnetic field in the plunging region is uniquely determined by the boundary conditions at the marginally stable circular orbit. There is also a minimum value of the energy that will allow a stable circular orbit. of the desired circular orbit? The relativistic central force orbits were previously studied in [2–5]. Transcribed image text: Find the condition for stable circular orbits for a potential energy of the form V(r) =- where ? In the test-mass limit, well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes. Gravitational force is ∝ r 2 1 , hence it decreases when r increases. See the answer See the answer See the answer done loading. Now the motion (when \( L_z > 0 \)) is much more interesting. (d) For a circular orbit at with radius a we need ∂U eff (r)/∂r| a = 0. In Schwarzschild metric, the orbital velocity for a circular orbit with radius $${\displaystyle r}$$ is given by the following formula: This region is bounded by the thick dark line (2S − x = 0) and the dashed line (S = 0). Express l c in terms of l, and express E c in terms of ì, l, and r p. ANS: From the circular orbit condition for the Kepler problem, the radius of a circular orbit is r c 2= l c /(ìk). It is convenient to consider the potential V̂ = V − Ẽ 2 /2 rather than V . 3.6.1. The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. _o P__ 9.0 8.0 7.0 6.0 5.C 4. Problem14. A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.. This relationship can be used to explore the behaviour of satellites in circular orbit around the Earth and make predictions about their speed, orbital radius and period of rotation. (313) or. The effective potential will have an extremum (local minimum or maximum) when d dr V eff = 0 ⇒ e−r/a r a 1+ r a = 2 b a = l2 mka (1) This equation can be written as f(x) = e−xx(1+x) = C with x = r/a and C a dimensionless constant C = l2/mka. 0 and the constant c is positive (negative) if ? orbit at a xed radius. 1. Orbital velocity in general relativity. In Schwarzschild metric, the orbital velocity for a circular orbit with radius r {displaystyle r} is given by the following formula: where r S = 2 G M c 2 {displaystyle scriptstyle r_{S}={frac {2GM}{c^{2}}}} is the Schwarzschild radius of the central body. These are compared with the circular orbit condition in the (post)-Newtonian approximation, hereafter PN, analysis of reference . (Keywords: builtin, built in, global functions.) And you can convince yourself that for r < ρ we get Feff > 0 and for r > ρ we get Feff < 0 so deviations from a circular orbit pull it back to a circular orbit. Show that if the electron is projected from a point with velocity and , then it will describe a circular orbit provided that. For circular orbits, the stationary states can be determined from a quantiza­ tion condition for the angular momentum, mvr = nh¯, n = 1, 2, 3, where r is the classical orbit, and v is the classical velocity for that orbit; the integer n is called the quantum number for that orbit. Timelike circular orbit exists in the light blue area. In principle, a circular orbit is a possible orbit for any attractive central force. However, not all forces result in stable circular orbits. Let us now consider the stability of circular orbits in a general central force-field. June 7, 2021. In this paper we investigate the innermost stable circular orbit (ISCO) (spin-aligned or anti-aligned orbit) for a classical spinning test particle with the pole-dipole approximation in the background of Kerr-Newman black hole in the equatorial plane. A brief description of … All wavelengths of the hydrogen spectrum are accurately predicted by the Bohr theory, which assumes that the single electron in hydrogen is held in stable, nonradiating circular orbits by the coulombic attraction of the stationary central proton of the hydrogen atom. Efforts to improve the stability of FAPbI 3 have focused on mixed cation-anion hybrid LHPs that incorporate several cations, anions, or both, such as the FA x MA 1– x double cation or the FA 1– x – y MA x Cs y triple cation. For what values of nis this circular orbit stable? In this case, we have 2d + R = (v 02 (d + R) /µ)/(1 + e cos π), which for v 0< v c gives a positive eccentricity. A powerful combination of spray, motion and noise not only acts as a deterrent to scare off animals, but also conditions them to stay away from the enforced area. We show that the stationary state of the magnetic field in the plunging region is uniquely determined by the boundary conditions at the marginally stable circular orbit. r about this radius. Proposition (a): The condition for a circular orbit, , under the force , is . Sketch them for the cases that n= 2, n= 1, and n= 7. We study the inner-most stable circular orbit (ISCO) of Kerr black hole in MOdified gravity (Kerr-MOG black hole) which is one of the exact solution of the field equation of modified gravity in strong gravity regime. It is defined mathematically as torsional stress divided by the torsional modulus of elasticity. Thus, Bohr's model of the atom postulates that electrons can only revolve in precise discrete orbits of specific radii, called 'stable orbits.' If ω2 > 0, the circular orbit is stable and the perturbation oscillates harmonically. In turns out that in this case, the orbit has a lower energy than the circular orbit, and, hence, the launch point is now the orbit’s apogee. Argue that if p n+ 2 is a rational number, these orbits are closed. Line 20: star 2’s momentum is opposite of star 1. Here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion. U ″ ( r 0) U ′ ( r 0) + 3 r 0 > 0, for stable orbits. we have a circular orbit for r = ρ we have Feff = 0. Use d2V eff(r)/dr 2 to demonstrate that we require n ≥ –2 for the circular orbit to be stable. What is the gravitational force exerted on the moon while it is in orbit around the planet? In the finite-mass case, there are a large variety of ways to define an ISCO in a post-Newtonian (PN) context. Abstract. Plugging this into Eqn (1) we obtain the following condition. ORBIT_YAW_BEHAVIOUR: 4: Orbits: Orbit around the centre point for this many radians (i.e. Assuming a power law, f = K r n, for the central force, solving it gives me the solution n > − 3. Here is the rest of the code where the stuff is actually calculated. Moreover, most of the solar system's large satellites, including the Earth's moon, rotate synchronously, keeping one … Built-in GDScript functions. the innermost stable circular orbit ~ISCO! 2 π r. 2 π r in one period T. Using the definition of speed, we have. [3] Problem 4 (Morin 7.4). The inner moons follow circular orbits around Jupiter interior to Io's orbit. Bound orbits which are not circular will oscillate around the radius of the stable circular orbit. Substituting in (2) gives the result (1). I've drawn three energy levels on the potential plot. In this paper we investigate the innermost stable circular orbit (ISCO) (spin-aligned or anti-aligned orbit) for a classical spinning test particle with the pole-dipole approximation in the background of Kerr-Newman black hole in the equatorial plane. The spectral lines of hydrogen are records of the wavelengths of radiations emitted or absorbed by hydrogen atoms. We focus on the plunging region between the black hole horizon and the marginally stable circular orbit in the equatorial plane, where the centrifugal force is unable to stably balance the gravitational force. 6.67\times 10^ {-11} \text { N} \cdot \text {m}^2 / \text {kg}^2 6.67×10−11 N ⋅m2/kg2. The solid line represents the location of circular orbits (stable and unstable). NaN: Use vehicle default, or current value if already orbiting. (The … Requirements for stable circular orbit with radius a for potential V(r) = kr n are: n > -2 , n ≠ 0, k = M 2 /(nma n+2). F or L → ∞, there exist a stable circular orbit at L 2. (a) If the central object is replaced with a more massive object, how must the orbital … The diagram shows a satellite orbiting the Earth. 1985-Fall-CM-U-3. If this condition holds, small radial deviations from the circular orbit will oscillate about r? Equation (12) of Mechanics-Notes 2 is (2) which is a basic equation of central force field. (9.27) 8.49) that any Kepler orbit can be written in the form r(˚) = Advance Condition will turn on auto advance when this condition is set. A special solution of the orbit equations is obtained if the potential has a minimum. Proposition (b): Further if the orbit is stable then (3) Therefore, according to Bohr's quantisation condition: mvr = nh/2π, where 'n' represents the principal quantum number of the orbit and is an integer (n = 1, 2, 3,…). If this condition holds, small radial deviations from the circular orbit will oscillate about r? (3-48)). 2Solution 2.1 Condition for a Central Orbit with Fixed Radius A condition for the existence of a central circular orbit with a fixed radius follows (for For instance, if you give it a nudge downward, the orbit will become a little elliptical, but it wouldn't fall. one described an electron as rnovins about the atomic nuclei in circular orbits such as shor.vn in Fig. Initial conditions: u(0) = 1/r min, 1/r max, u0(0) = 0. In Sec. Let the angular momentum be L. (a)Find the radius r 0 of the circular orbit. This problem has been solved! Initial conditions: u(0) = 1/r min, 1/r max, u0(0) = 0. 1.) For an object to remain in a steady, circular orbit it must be travelling at the right speed. Math functions and other utilities. 55 with map energy|to forecast circular orbits. Substituting into the above stability criterion, we obtain. Does this One of the most strong rea-sons is that in many sources, the maximum frequency of the kHz QPOs is in narrow range between 1.1 and 1.2 kHz, although they are thought to have very different mass accre-tion rates and magnetic fields. Just for reference, the eccentricity of Earth's orbit is .0167. Even though Earth’s orbit is very nearly circular, the intensity of sunlight falling on a given location on the planet’s surface changes as it orbits around the sun. The moon has a mass of 1×1022 kg, and the gravitational field strength at a distance R from the planet is 0.001 N/kg. In that paper a search was made for the inner most stable circular orbit in the absence of radiation reaction terms in the equations of motion. III is devoted to the study of circular orbits around a rotating black hole. In the finite-mass case, there are a large variety of ways to define an ISCO in a post-Newtonian (PN) context. (c) E = T + U = ½m (dr/dt) 2 + M 2 / (2mr 2) + kr n = ½m (dr/dt) 2 + U eff (r). Bertrand's Theoremcharacterizes the force laws that govern stable circular orbits. It states that the only force laws permissible are the Hooke's Potential and Inverse Square Law. The proof of the theorem involves some perturbation techniques and series expansion. where v is the linear speed. Earth and the moon, sun, and planets have predictable patterns of movement. mechanics - mechanics - Circular orbits: The detailed behaviour of real orbits is the concern of celestial mechanics (see the article celestial mechanics). Barack and Sago have recently computed the shift of the innermost stable circular orbit (ISCO) of the Schwarzschild spacetime due to the conservative self-force that arises from the finite-mass of … The angular momentum of the electron is quantized according to the expression: L = mvr = n ħ List of core built-in GDScript functions. Do your sketches confirm The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. Gravity provides the force needed to maintain stable orbit of planets around a star and also of moons and artificial satellites around a planet. Necessary centripetal force to keep the satellite orbiting in a stable circular orbit is provided by the force of gravitational attraction between the earth and the satellite. Like the orbital integral, the orbital differential equation describes the rela-tion between the radial and angular coordinates of an orbit, a relation from (c) For the stable case, show that the period of small oscillations about the circular orbit is ˝ osc = ˝ orb= p n+ 2. Acceleration and Circular Motion When an object moves in a circular orbit, the direction of the velocity changes and the speed may change as well. These are compared with the circular orbit condition in the (post)-Newtonian approximation, hereafter PN, analysis of reference . Express l c in terms of l, and express E c in terms of ì, l, and r p. ANS: From the circular orbit condition for the Kepler problem, the radius of a circular orbit is r c 2= l c /(ìk). A particle of mass mmoves in a potential V(r) = rk. For what values of n is this circular orbit stable? The potential wells of black holes are so deep and steep, that they attract matter from their entire neighborhood. 6.67 × 1 0 − 11 N ⋅ m 2 / kg 2. Stable circular orbit at r = Unstable circular orbit at r = Analogous to conditions for stable & unstable equilibrium in static equilibrium problems! Read more. The DCS381 Reciprocating Saw with keyless blade clamp allows for quick blade changes. To find the period of a circular orbit, we note that the satellite travels the circumference of the orbit. The movement is due to the electrostatic attraction that the nucleus exerts on it. r 0 3 = L 2 μ U ′ ( r 0). The requirement that the circular orbit be stable for all radii already restricts the form of the force law through the inequality condition of P2 > 0 (Eq. Muller (1994), for example, collected data on the inclination of the Earth’s orbit over a 100,000 year cycle, correlated it with the occurrence of ice ages, ruled out the plausibility of orbital eccentricity as a cause for the occurrence of ice ages, and inferred that the bounce in the Earth’s orbit likely caused the ice … If a satellite is in a perfectly circular orbit around its primary, it keeps the same face toward its parent planet, and this tidal bulge remains fixed relative to the planet. In order for a circular orbit to occur, the orbiting object has to achieve the right velocity, and the …

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