Sub-Keplerian Orbits

Updated 27 July 2025

Sub-Keplerian orbit is an orbital regime where the rotational velocity of material is lower than the Keplerian value due to forces like pressure gradients, magnetic fields, and radiation.
This flow paradigm leads to disks with steeper density profiles, altered temperature structures, and reduced crystalline silicate fractions in inner regions.
Improved modeling of sub-Keplerian dynamics advances our understanding of disk evolution, planet formation, and accretion processes around stars and compact objects.

A sub-Keplerian orbit refers to an orbital or accretion flow regime in which the rotational velocity or angular momentum of material is systematically below the value required for a classical, circular Keplerian orbit at a given radius from the central object. This concept arises in a range of astrophysical contexts, including circumstellar disks, planetary systems, accretion onto black holes, and circumstellar environments of evolved stars. Sub-Keplerian regimes are of particular importance in disk evolution, accretion dynamics, and disk chemistry, where deviations from idealized Keplerian motion induce significant structural, kinematic, and compositional effects.

1. Formal Definition and Theoretical Basis

In the Keplerian regime, an object of negligible mass orbits a central body under gravity with a velocity $v_\mathrm{Kep} = \sqrt{GM_*/R}$ , where $M_*$ is the central mass and $R$ is the distance to the center. In contrast, a sub-Keplerian orbit is characterized by $v_\phi < v_\mathrm{Kep}$ , so the material is supported against gravity by a combination of reduced centrifugal force and additional forces or dynamical processes.

Formally, the deviation is often quantified as the difference between the azimuthal velocity of the flow ( $\Omega$ or $v_\phi$ ) and the Keplerian value ( $K$ or $\Omega_\mathrm{K}$ ):

$v_\phi^2 = \frac{GM_*}{R} + \frac{R}{\rho}\frac{dP}{dR} + \cdots$

where the $dP/dR$ term accounts for radial pressure gradients, and additional terms can represent the effect of radiation pressure, magnetic fields, or mass loading.

Sub-Keplerian flow frequently occurs during accretion from an external envelope or infalling cloud, shock-dominated black hole accretion, and in the presence of substantial non-gravitational forces (e.g., radiation or magnetic stresses).

2. Sub-Keplerian Accretion in Disk Evolution Models

The evolution of disks with sub-Keplerian accretion is particularly important in models of circumstellar disk formation. In one-dimensional (1D) or semi-2D viscous disk models, the infall of envelope material at sub-Keplerian velocities presents a challenge for angular momentum conservation, as standard treatments tend to deposit infalling matter at the radius it would attain if it were rotating at the Keplerian rate.

Recent models rigorously account for these effects by modifying the radial flow equation. The updated disk evolution equation, derived from angular momentum conservation, introduces a correction term proportional to the difference between the local Keplerian angular velocity ( $K$ ) and the actual rotation rate of the infalling material ( $\Omega$ ):

$u_R(R,t) = -\frac{3}{\Sigma\sqrt{R}} \frac{\partial}{\partial R}(\Sigma \nu \sqrt{R}) - \frac{2R S}{\Sigma} \frac{K - \Omega}{K}$

Here:

$\Sigma$ is the surface density,
$\nu$ is the kinematic viscosity,
$S$ is the mass infall rate per unit area,
$K$ is the Keplerian angular frequency, and
$\Omega$ is the actual rotation frequency of the infalling gas.

This additional term represents the angular momentum deficit of material accreting with $v_\phi < v_\mathrm{Kep}$ and operates most strongly in the disk's outer regions, steepening the surface density profile and limiting the radial expansion of the disk (1005.1261).

3. Effects of Disk Vertical Structure and Infall Geometry

A critical advance in the treatment of sub-Keplerian accretion is the explicit inclusion of the disk's vertical extent. Rather than assuming accretion directly onto the midplane, ballistic calculations show that material with sub-Keplerian specific angular momentum lands on the disk’s surface layers, with the specific radius set by the intersection of the infalling trajectory with the disk surface (not the midplane). This is governed by:

$\rho_\mathrm{env} u^2 = \frac{k T_s \rho_\mathrm{disk}}{\mu m_p}$

where $\rho_\mathrm{env}$ is the envelope density, $u$ the infall velocity, $T_s$ the disk surface temperature, and $\mu m_p$ the mean particle mass.

The practical result is that the infalling mass is deposited over a broader range of radii. Significantly, only a minority (about one-third) of the envelope mass reaches the hot, inner disk where temperatures are sufficient to anneal amorphous silicate grains into crystalline form (1005.1261). The remainder is deposited at larger radii, yielding a more compact disk with a steeper surface density gradient.

4. Consequences for Disk Properties, Composition, and Thermal Processing

The altered distribution of angular momentum and deposition radii in sub-Keplerian accretion flows imposes a series of measurable consequences on disk properties:

Disk size and mass distribution: Disks formed by sub-Keplerian accretion are systematically smaller, with more mass concentrated at smaller radii and a steeper surface density profile.
Temperature structure: The inner regions are slightly cooler due to increased density, while outer regions can be warmer due to enhanced irradiation.
Chemical composition: The warmer outer disk reduces the retention of volatile ices (e.g., solid CO) relative to models assuming pure Keplerian accretion.
Dust processing and crystallinity: Only ~36% of the accreted material is annealed to crystalline silicates, substantially less than predicted under flat-disk infall assumptions. This aligns predicted crystalline fractions (1–30%) much more closely with those observed in protoplanetary disks. The post-annealing radial mixing is then essential to distribute crystalline material to large radii.

The table below summarizes some key differences between sub-Keplerian updated (SK) and classic Keplerian (Kep) deposition treatments:

Property	Sub-Keplerian (SK)	Keplerian (Kep)
Disk size	Smaller, sharper edge	Larger, spreads outward
Crystalline silicate frac.	1–15% (inner 10–30 AU)	Near 100% (hot inner disk)
Temperature (outer disk)	Few degrees warmer	Slightly cooler

5. Broader Implications for Disk Evolution and Planet Formation

The inclusion of sub-Keplerian effects has substantial consequences for disk evolution theory:

Angular momentum conservation: The updated equations conserve angular momentum exactly, removing artificial jumps or discontinuities in mass profiles.
Model–observation alignment: Predictions for the size, temperature, and crystalline content of disks are brought into closer correspondence with observations.
Planet formation: The altered distribution of solids (e.g., location, concentration, and crystal fraction) affects the initial conditions for planetesimal and planetary growth.
Ices and chemistry: Local temperature modifications alter volatile partitioning, affecting molecular abundances and ice/rock ratios at different disk radii.

The updated treatment also provides a more general template for incorporating infall at non-Keplerian velocities into semianalytic disk models, which remain important for evolutionary and population studies (1005.1261).

While the detailed mechanisms considered above are specific to protostellar disks, sub-Keplerian flows are ubiquitous in astrophysical environments:

Accretion onto compact objects: Low angular momentum flows around black holes or neutron stars routinely develop sub-Keplerian transonic regions, with important implications for the development of standing or oscillating shocks, as well as for the two-component structure of disks (Keplerian + sub-Keplerian) (Chakrabarti, 18 Sep 2024).
Disks influenced by non-gravitational forces: In evolved star environments, radiation pressure on dust can induce sub-Keplerian rotation by counteracting gravity, with the observed velocity lag constrained by dust properties and the local SED (Haworth et al., 2017).
Circumbinary environments and perturbed planetary systems: Gravitational perturbations from non-axisymmetric potentials lead to orbits that cannot be described by simple Keplerian ellipses; the sub-Keplerian terminology is sometimes extended to cover such non-Keplerian or complex time-dependent systematics (1212.2545).

A plausible implication is that rigorous sub-Keplerian treatments, analogous to those now used for envelope–disk infall, will be necessary to improve the realism of models across a wide range of disk- and flow-dominated systems.

7. Concluding Remarks and Research Directions

The rigorous treatment of sub-Keplerian orbits in disk evolution models represents a significant advance in the quantitative modeling of disk structure and evolution. The primary conclusions are:

The inclusion of sub-Keplerian correction terms, as in Eq. (2) above, is necessary for exact angular momentum conservation in semianalytic models.
Modifying both the kinematic infall prescription and the vertical structure of disk models produces disks with properties in much closer agreement with observed distributions of size, crystalline content, and chemical makeup.
The re-distributed deposition of mass and angular momentum sets initial conditions for subsequent planet formation and can be expected to affect both the migration rates and chemical compositions of forming planetary bodies.

Open research directions include the exploration of sub-Keplerian infall in disks at later evolutionary stages, the paper of the interplay between sub-Keplerian accretion, photoevaporation, and turbulence, and the application of these methods to population syntheses for planet-forming environments (1005.1261).