Satellite Motion in a Manev Potential with Drag

In this paper, we consider a satellite orbiting in a Manev gravitational potential under the influence of an atmospheric drag force that varies with the square of velocity. Using an exponential atmosphere that varies with the orbital altitude of the satellite, we examine a circular orbit scenario. In particular, we derive expressions for the change in satellite radial distance as a function of the drag force parameters and obtain numerical results. The Manev potential is an alternative to the Newtonian potential that has a wide variety of applications, in astronomy, astrophysics, space dynamics, classical physics, mechanics, and even atomic physics.

💡 Research Summary

The paper investigates the orbital evolution of an artificial satellite that moves under a modified gravitational field known as the Maneuver (Manev) potential while simultaneously experiencing atmospheric drag that scales with the square of the satellite’s velocity. The authors adopt a circular‑orbit approximation and model the Earth’s atmosphere as an exponential function of altitude, which is a standard representation for low‑Earth‑orbit (LEO) environments. Their primary goal is to derive an analytical expression for the rate of change of the orbital radius (dr/dt) that explicitly incorporates both the Manev correction to gravity and the drag force, and then to explore how the Manev parameter influences the decay rate relative to the classical Newtonian case.

Theoretical framework
The Manev potential is defined as

 V_M(r) = –GM/r + β/r²,

where G is the gravitational constant, M is Earth’s mass, r is the radial distance from Earth’s centre, and β is a constant that quantifies the strength of the 1/r² correction. When β > 0 the effective gravity is stronger than Newton’s law, while β < 0 weakens it. This potential has been employed in various contexts—planetary perihelion precession, relativistic corrections, and even in atomic‑scale analogues—because it introduces a simple, analytically tractable deviation from the inverse‑square law.

Atmospheric density is taken as

 ρ(h) = ρ₀ exp(–h/H),

with ρ₀ the sea‑level density, H the scale height (≈ 8.5 km), and h = r – R_E the geometric altitude above Earth’s mean radius R_E. The drag force is modeled in the usual quadratic form

 F_D = ½ C_D A ρ v²,

directed opposite to the instantaneous velocity vector v. Here C_D is the drag coefficient, A the reference cross‑sectional area, and v the orbital speed.

Derivation of the radial decay law
Assuming a perfectly circular orbit of radius r₀, the centripetal balance in the presence of the Manev correction reads

 v²/r = GM/r² – 2β/r³.

Solving for v gives a slightly higher speed than the Newtonian circular speed when β > 0. The mechanical energy per unit mass is

 E = ½ v² + V_M(r).

The rate of energy loss due to drag is

 dE/dt = –F_D v = –½ C_D A ρ v³.

Differentiating E with respect to time and substituting the expression for v from the balance equation yields a first‑order differential equation for r(t). After algebraic manipulation the authors obtain

 dr/dt = –