Acknowledgement. We thank Science for their permission to use an excerpt from:
Broadfoot, A. L., et al. 1986. Ultraviolet Spectrometer observations at Uranus. Science 233 (4759), 74-79. (Excerpt from pp. 77-78.)
Exospheric drag and ring evolution. The extended Uranian exosphere influences the dynamical evolution of the rings. Even in the absence of collisions, the inner part of an exosphere should be in solid-body rotation (20), a result verified by our own numerical calculations. The reduction of effective gravity toward the equatorial plane because of planetary spin also increases the scale height and therefore the density at high altitudes. In the region of the rings (1.6 to 2 R_U), this model is well fitted by the extrapolation expression
n_H = 7e-6 e^(32.4/r) cm^(-3) 
where r is in units of the planetary radius (26,200 km); n_H is normalized to our occultation value at 27,930 km. The drag force can be expressed as D=C_D A_p rho V_rel^2 for a ring particle moving with relative velocity V_rel through a gas of density rho. Here C_D is the drag coeffficient, usually near unity, and A_p is the mean cross-sectional area of the particle. Equating the drag torque to the loss of angular momentum of an orbiting particle, we can derive the decay rate of the orbit. For a ring particle of mass M_p in a circular orbit in a hydrogen exosphere of number density in n_H, this rate is given by
dr/dt = -2 C_D (A_p/M_p) n_H m_H r V_rel^2/V_0 
where m_H is the mass of the hydrogen atom and V_0 is the orbital velocity of the particle. Assuming a corotating exosphere whose density is given by Eq. 1 and considering spherical particles of radius a (in centimeters) and density 1 cm^(-3), we obtain approximate orbital decay rates of
dr/dt ~= -5e-16 a^(-1) r^1/2 e^32.4/r 
(in units of Uranus radii per year).
Numerical integration of Eq. 2 leads to the orbital lifetimes as a function of planetocentric distance and particle radius shown in Fig. 6 Orbital lifetimes are proportional to particle radius, and small particles will be selectively depleted. Such short orbital lifetimes have important implications for the population of small particles in the rings. The forces that confine the narrow, high optical depth, rings of Uranus are primarily gravitational. A strong nongravitational force will truncate the original size distribution of the rings at some minimum particle size, biasing the size distribution toward larger particles. An extended complex of tenuous rings, seen in the high phase angle ring images (21), indicates that a population of small (~1 micrometer) particles is present within the ring system. Given the lifetime (1e2 to 1e3 years) of such particles, this may represent a steady-state population of fine material derived from the rings or from shepherd satellites (or from both) and spiraling toward the planet.
A ring experiences a collective drag torque from the smallest particles remaining in the ring. The magnitude of this torque depends on the particle size distribution, which for the Uranian rings is not well known. However, estimates of decay time are possible for the alpha, beta, and epsilon rings where surface mass densities and mean optical depths are known. If each ring is considered an isolated system (that is, disregarding the contribution of possible shepherd satellites), then the drag torque on a ring is proportional to the ratio of the weighted average of particle areas divided by the weighted average of particle masses. For each of these three rings (assuming a single particle size), this can be approximated by tau/SIGMA, where tau is the mean optical depth and SIGMA is the surface mass density of the ring. From the observed ring parameters (22), we obtain lifetimes of 4e6, 8e6, and 6e8 years for the alpha, beta, and epsilon rings, respectively. Although orbital lifetimes would be increased by additional mass in ring shepherds or by a substantial population of large particles, lifetimes still would not be indefinite, and a recent origin of the rings should be considered.
It is legitimate to ask how confidently we can extrapolate the H number density at 1.2 R_U to the distance of the rings. Of the measured quantities, density and temperature, the latter contributes nearly all the uncertainty. The 750 K temperature is based on the fit in Fig. 1A rather than on the altitude difference between A and B. We estimate an uncertainty of +/-100 K. The corresponding uncertainties in H number density and orbital lifetimes are shown in Fig. 6. The effect of the ring drag on the gas is negligible because nearly all the atoms are launched on ballistic trajectories from the exobase, with a free-fall time of 1 hour or less.
Last updated Feb-27-1997