Analytic gravitational-force calculations for models of the Kuiper Belt, with application to the Pioneer anomaly

We use analytic techniques to study the gravitational force that would be produced by different Kuiper-Belt mass distributions. In particular, we study the 3-dimensional rings (and wedge) whose densities vary as the inverse of the distance, as a constant, as the inverse-squared of the distance, as well as that which varies according to the Boss-Peale model. These analytic calculations yield physical insight into the physics of the problem. They also verify that physically viable models of this type can produce neither the magnitude nor the constancy of the Pioneer anomaly.

Figure 1

A plot of $-a_{shell}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs $r$ in AU.

Figure 2

A plot of $-a_{T/p}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs $r$ in AU.

Figure 3

A plot (solid line) of $-a_{BP}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs $r$ in AU. Also shown is a dashed plot of ${\rho }_{BP}(p)$ in units of ${10}^{-15}\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$.

Figure 4

A plot of $-a_{1/p}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs distance in AU, obtained from a $(1/p)$-density ring, with a 3-dimensional calculation.

Figure 5

A plot of $-a_{Con}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs distance in AU for a uniform ring.

Figure 6

A plot of $-a_{1/r^2}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs $r$ in AU.

Figure 7

A plot of $-a_{T/p^2}(r)$ in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$ vs $r$ in AU.

Figure 8

Plots of $[-a(r)]$, in units of ${10}^{-8}\mathrm{cm}/{\mathrm{s}}^2$, vs distance (in AU), for: (i) the 3D, $1/p$-density ring, $[-a_{1/p}(r)]$—short-dashed line, (ii) the Boss-Peale 2D “thin ring,” $[-a_{BP}(r)]$—narrow line, (iii) the constant-density ring, $[-a_{Con}(r)]$—medium line, (iv) the $1/r^2$-density, wedge, $[-a_{1/r^2}(r)]$—dashed line, and (v) the $1/p^2$-density, “thin ring,” $[-a_{T/p^2}(r)]$—wide line.