Yarkovsky effect

The Yarkovky effect is implemented according to [VokrouhlickyFarinella99] and [VokrouhlickyMilaniChesley00] and is available in the schemes (we recommend scheme 1):

  • Velocity kick \(\mathbf{a_Y}\)
    set Use Yarkovsky = 1 in the param.dat file.
    See Equation (7) and (8)
  • Time averaged change in sami-major axis \(\frac{da}{dt}\)
    set Use Yarkovsky = 2 in the param.dat file.
    See Equation (9) and (10)
The following parameters are relevant for the Yarkovsky effect and can be set in the param.dat file:

  • Use Yarkovsky

  • Solar Constant: Solar Constant at 1 AU in W / \(\text{m}^2\)

  • Asteroid eps: Emissivity factor

  • Asteroid rho: density of the body in kg/ \(\text{m}^3\)

  • Asteroid C: Specific Heat Capacity in \(\text{J} \, \text{kg}^{-1} \text{K}^{-1}\)

  • Asteroid A: Bond albedo

  • Asteroid K: Thermal conductivity in \(\text{W} \, \text{m}^{-1} \text{K}^{-1}\)

Note that the calculation of the Yarkovsky effect uses the Asteroid rho value for the calculation of the thermal inertia \(\Gamma\) and not the individual particle densities.

(7)\[\mathbf{a}_{seasonal} = \frac{4}{9}\frac{(1 - A) \Phi}{(1 + \lambda)} \times \sum_{k \ge 1} G_k\left[s_P \alpha _k \cos (knt + \delta_k) + s_Q \beta_k \sin(knt + \delta_k) \right] \mathbf{s},\]
(8)\[\mathbf{a}_{diurnal} = \frac{4}{9}\frac{(1 - A) \Phi}{(1 + \lambda)} G\left[\sin \delta + \cos \delta \mathbf{s} \times \right] \frac{\mathbf{r} \times \mathbf{s}}{r}.\]
(9)\[\left(\frac{da}{dt} \right)_{diurnal} = - \frac{8}{9} \frac{(1-A)\Phi}{n} \frac{G \sin \delta}{1 + \lambda} \cos \gamma\]
(10)\[\left(\frac{da}{dt} \right)_{seasonal} = \frac{4}{9} \frac{(1-A)\Phi}{n} \sum_{k \ge 1} \frac{G_k \sin \delta_k}{k} \chi_k \bar{\chi_k},\]

Set the call interval

With the parameter Yarkovky Inteval the calling interval of the Yarkovsky function can be set. This means that the function is called fewer times, and in increased drift rate \(dx = a * dt * YarkovskyInterval\). With the time averaged Yarkovsky function (mode 2) it is possible to use quite large interval numbers. The direct Yarkovsky effect (mode 1) is more sensitive to values greater than \(\approx 100\). Strictly speaking ,this argument violets the symplectic nature of the integrator, but as long as the Yarkovky force is small it is OK.

Test of the Yarkovsky effet

In Fig. 13 is shown a test of the Yarkovsky effect following [FarinellaVokrouhlickyHartmann98] and [BRB00].

Relevant parameters for this example:

  • Use Yarkovsky = 1 (2)

  • Asteroid emissivity eps = 1.0

  • Asteroid density rho = 3500.0

  • Asteroid specific heat capacity C = 680.0

  • Asteroid albedo A = 0.0

  • Asteroid thermal conductivity K = 2.65

  • Solar Constant = 1367

Initial conditions:

  • Semi-major axis a = 2 AU

  • Eccentricity = 0

  • Inclination = 0

  • Argument of perihelion = 0 - 2 \(\pi\)

  • Longitude of ascending node = 0 - 2 \(\pi\)

  • Mean anomaly = 0 - 2 \(\pi\)

  • Density \(\rho\) = 3500.0 kg/m^3

  • Physical radius R = 0.1 - 100 m

  • Rotation frequency \(\omega\) = 5 h

  • Obliquity \(\gamma_{\text{Seasonal}}\) = \(90^\circ\)

  • Obliquity \(\gamma_{\text{Diurnal}}\) = 0

_images/Yarkovsky.png

Fig. 13 Drift rate of the seasonal and diurnal Yarkovsky effect. Computed after 10000 years and averaged over 1 orbit.