Dynamical Models

DiffOrb propagates one target small body in the Barycentric Celestial Reference System (BCRS) as a function of Barycentric Dynamical Time (TDB). A dynamical model gives the acceleration used by that propagation.

Core Model

DiffOrb only integrates the target small body. It does not integrate the Sun, planets, Moon, Pluto, or massive asteroid perturbers. Those perturbing bodies are read from SPK-format ephemerides.

Ephemeris-Backed Bodies

DiffOrb can load one SPK kernel or a list of SPK kernels. The kernels may store segments with different centers. DiffOrb finds and combines the needed path segments so a body state can be evaluated with respect to the Solar System Barycenter.

Many DE and asteroid SPK kernels store Chebyshev coefficients for position. DiffOrb evaluates the position at the requested TDB epoch. It then obtains velocity and acceleration from the same position interpolation by automatic differentiation. This keeps position, velocity, and acceleration tied to one interpolation model.

Gravitational Terms

DiffOrb supports additive gravitational force terms.¹

• Newtonian point-mass gravity gives the inverse-square acceleration from selected bodies.
• Point-mass parameterized post-Newtonian (PPN) gravity adds the relativistic correction used for Solar System propagation.
• The second zonal harmonic (J2) models the leading oblateness effect of the Sun or Earth.

Users can choose which bodies use Newtonian gravity and which bodies use PPN gravity. Common setups use PPN for the Sun and major planets, and Newtonian gravity for selected massive asteroids.

Non-Gravitational Terms

For small bodies, DiffOrb uses empirical non-gravitational acceleration models for cometary outgassing, the Yarkovsky effect, and radiation pressure.²³⁴ These models are written in one heliocentric radial-transverse-normal (RTN) form:

\[ \mathbf{a}_{\mathrm{ng}} = g(r)\left(A_1 \hat{\mathbf{r}} + A_2 \hat{\mathbf{t}} + A_3 \hat{\mathbf{n}}\right). \]

Here \( r \) is the heliocentric distance. The unit vectors \( \hat{\mathbf{r}} \), \( \hat{\mathbf{t}} \), and \( \hat{\mathbf{n}} \) are the orbital radial, transverse, and normal directions. The parameters \( A_i \) \((i=1,2,3)\) are fitted acceleration parameters at a heliocentric distance of 1 au. A1 is radial, A2 is transverse, and A3 is normal. \( g(r) \) is a heliocentric-distance law.

For cometary outgassing, DiffOrb uses the Marsden cometary outgassing model.² In this model, \( g(r) \) represents the sublimation rate as a function of heliocentric distance. The default \( g(r) \) corresponds to water-ice sublimation. Its shape parameters can be changed to represent sublimation processes of other materials, such as sodium or forsterite sublimation.⁵⁶

By setting

\[ g(r)=\left({1\,\mathrm{au}\over r}\right)^2, \]

the same RTN form can model the Yarkovsky effect as the purely transverse acceleration \( A_2 g(r) \), and radiation pressure as the purely radial acceleration \( A_1 g(r) \).⁷

These are empirical acceleration models for orbit determination. They are not complete physical models of gas flow, thermal emission, or surface scattering.

Model Boundary

The force model defines acceleration. It does not define observations. Optical and radar observations use the propagated target state, site states, and light-time models in later layers.

The force model also does not choose the numerical method. That belongs to the integrator layer.

Read Next

• Read Numerical Integrators And Dense Trajectories for the propagation layer that uses the dynamical model.
• Read Light-Time Model for one-way optical light time, two-way radar light time, and radar Doppler.
• Use Configure Force Models And Dynamic Systems when you need the concrete force-model setup.
• Use the Dynamics API for symbol-level details.

References

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1. Urban, S. E., & Seidelmann, P. K. (eds.). Explanatory Supplement to the Astronomical Almanac, especially the sections on Solar System equations of motion. ↩

2. Marsden, B. G., Sekanina, Z., & Yeomans, D. K. (1973). Comets and nongravitational forces. V. The Astronomical Journal, 78, 211-225. https://doi.org/10.1086/111402 ↩↩

3. Vokrouhlicky, D., Bottke, W. F., Chesley, S. R., Scheeres, D. J., & Statler, T. S. (2015). The Yarkovsky and YORP Effects. In P. Michel, F. DeMeo, & W. Bottke (eds.), Asteroids IV, 509-531. https://doi.org/10.2458/azu_uapress_9780816532131-ch027 ↩

4. Vokrouhlicky, D., & Milani, A. (2000). Direct solar radiation pressure on the orbits of small near-Earth asteroids: observable effects? Astronomy and Astrophysics, 362, 746-755. ↩

5. Sekanina, Z., & Kracht, R. (2015). Strong Erosion-Driven Nongravitational Effects in Orbital Motions of the Kreutz Sungrazing System's Dwarf Comets. The Astrophysical Journal, 801, 135. https://doi.org/10.1088/0004-637X/801/2/135 ↩

6. Sekanina, Z., & Kracht, R. (2014). Disintegration of Comet C/2012 S1 (ISON) Shortly Before Perihelion: Evidence from Independent Data Sets. arXiv. https://doi.org/10.48550/arXiv.1404.5968 ↩

7. Farnocchia, D., Chesley, S. R., Vokrouhlicky, D., Milani, A., Spoto, F., & Bottke, W. F. (2013). Near Earth Asteroids with measurable Yarkovsky effect. Icarus, 224(1), 1-13. https://doi.org/10.1016/j.icarus.2013.02.004 ↩