A Planet or Primordial Black Hole in the Outer Region of the Solar System and the Dust Flow near Earth’s Orbit

Abstract

In recent years, evidence has been obtained that in the outer region of the Solar System (in the inner Oort cloud) at a distance of ~300–700 AU from the Sun, there may be a captured planet or a primordial black hole. In this paper, we show that the gravitational scattering on this object of dust particles located in the same region can transfer them to new elongated orbits reaching the Earth’s orbit. With the mass of the captured object on the order of 5–10 Earth masses, the calculated dust flow near the Earth is ~0.1–3 µg m^–2 year^–1 is comparable in order of magnitude with the observed flow. This effect gives a joint restriction on the parameters of the captured object and on the amount of dust in the Oort cloud.

APPENDIX

1.1 SIMPLE ESTIMATES

To study the scattering of dust particles by the gravitational field of the PBH, we performed a numerical calculation. In this section, we will show how the dust flux can be estimated by an order of magnitude using a simple approach. Let the characteristic radius of the dust cloud be ${{r}_{i}}$, then its density is

$${{\rho }_{d}} \simeq \frac{{3{{M}_{d}}}}{{4\pi r_{i}^{3}}}{\kern 1pt} .$$

(A.1)

Let us consider a dust particle moving along an orbit around the Sun at a distance ${{r}_{i}}$ with a velocity ${{{v}}_{i}} \simeq {{(G{{M}_{ \odot }}{\text{/}}{{r}_{i}})}^{{1/2}}}$, and suppose that a PBH with a mass ${{M}_{{{\text{PBH}}}}}$ passes by the dust particle at a minimum distance of $b$. We assume that the initial orbits of both the dust particle and the PBH are not circular but not highly elongated, with an eccentricity of $(1 - e) \sim 1$. In this case, the relative velocity of the PBH and the dust particle is ${{{v}}_{{{\text{rel}}}}} \sim {{{v}}_{i}}$. When the dust particle is passed by the PBH, it gains a velocity boost toward the point of closest approach

$$\Delta {v} \simeq \frac{{2G{{M}_{{{\text{PBH}}}}}}}{{b{\kern 1pt} {{v}_{{{\text{rel}}}}}}}{\kern 1pt} .$$

(A.2)

This expression represents the limiting form of the impulse approximation, which is valid here to an order of magnitude. The velocity of the dust particle after the interaction is denoted by ${{{v}}_{f}}$. If the velocity boost largely compensates for the initial velocity, the dust particle loses angular momentum and begins to fall almost directly towards the Sun on an elongated orbit. The condition for the dust particle to enter inside the Earth’s orbit when it approaches the Sun can be found from the conservation of angular momentum:

$$l \sim {{r}_{i}}{\kern 1pt} {{{v}}_{f}} \sim {{{v}}_{{\text{E}}}}{{R}_{{\text{E}}}},$$

(A.3)

where ${{R}_{{\text{E}}}} = 1$ AU is the radius of Earth’s orbit and the velocity ${{{v}}_{{\text{E}}}} \simeq {{(G{{M}_{ \odot }}{\text{/}}{{R}_{{\text{E}}}})}^{{1/2}}}$. From (A.3) we obtain

$${{{v}}_{f}} \sim {{{v}}_{i}}{{\left( {\frac{{{{R}_{{\text{E}}}}}}{{{{r}_{i}}}}} \right)}^{{1/2}}} \ll {{{v}}_{i}}.$$

(A.4)

From the condition of compensating the initial velocity of the dust particle $\Delta {v} \sim {{{v}}_{i}}$ we obtain

$$b \simeq \frac{{2G{{M}_{{{\text{PBH}}}}}}}{{{v}_{i}^{2}}}{\kern 1pt} .$$

(A.5)

For typical parameters, the $b$ value is on the order of 3 million km. As can be seen from (A.2), the final velocity of the dust particle will be ${{{v}}_{f}}$ only for a certain narrow interval of the impact parameters:

$$\frac{{\delta b}}{b} \simeq \frac{{{{{v}}_{f}}}}{{{{{v}}_{i}}}}{\kern 1pt} .$$

(A.6)

If the initial velocity of the dust particle was directed outside the plane perpendicular to the trajectory of the PBH, then after compensating the velocity ${{v}_{i}}$ the dust particle will gain additional velocity perpendicular to the radius vector directed away from the Sun. For the dust particle to still be able to enter within Earth’s orbit, this additional velocity should not exceed ${{v}_{f}}$. It follows that only dust particles with initial velocities within an angle $\phi \simeq {{v}_{f}}{\text{/}}{{v}_{i}}$ relative to the specified plane can enter inside Earth’s orbit. On the contrary, the gain in velocity along the radius vector will not affect the entry into Earth’s orbit and will not change the order of magnitude.

Summarizing the above, we obtain that the mass of dust directed inside Earth’s orbit per unit of time during scattering by the PBH is

$$\frac{{d{{M}_{d}}}}{{dt}} \simeq b{\kern 1pt} (\delta b){\kern 1pt} \phi {\kern 1pt} {{\rho }_{d}}{\kern 1pt} {{{v}}_{i}}.$$

(A.7)

Let us consider the dust particles that start to pass within Earth’s orbit after scattering on the PBH. Let us denote the full orbital period of a dust particle after scattering as T and the time it spends inside Earth’s orbit as Δt. The probability of a dust particle being inside Earth’s orbit can be estimated as follows

$${{P}_{t}} = \frac{{\Delta t}}{T} \sim \frac{{(2{{R}_{{\text{E}}}}{\text{/}}{{{v}}_{{\text{E}}}})}}{{(2{{r}_{i}}{\text{/}}{{{v}}_{i}})}}{\kern 1pt} .$$

(A.8)

Considering the time of dust particle ejection from the orbit t_ej, the steady-state dust density inside Earth’s orbit is

$${{\rho }_{s}} \sim \frac{{{{P}_{t}}{\kern 1pt} (d{{M}_{d}}{\text{/}}dt){\kern 1pt} {{t}_{{{\text{ej}}}}}}}{{(4\pi R_{{\text{E}}}^{3}{\text{/}}3)}}{\kern 1pt} ,$$

(A.9)

and its flux is

$$F = {{\rho }_{s}}{{{v}}_{{\text{E}}}} \simeq \frac{9}{{4{{\pi }^{2}}}}{\kern 1pt} \frac{{GM_{{{\text{PBH}}}}^{2}{{M}_{d}}{\kern 1pt} {{t}_{{{\text{ej}}}}}}}{{{{M}_{ \odot }}{{R}_{{\text{E}}}}r_{i}^{4}}}{\kern 1pt} .$$

(A.10)

Numerically, we obtain

$$\begin{gathered} F \simeq 0.4{{\left( {\frac{{{{M}_{{{\text{PBH}}}}}}}{{10{{M}_{ \oplus }}}}} \right)}^{2}}\left( {\frac{{{{M}_{d}}}}{{5{{M}_{ \oplus }}}}} \right)\left( {\frac{{{{t}_{{{\text{ej}}}}}}}{{2.5 \times {{{10}}^{6}}\;{\text{years}}}}} \right) \\ \, \times {{\left( {\frac{{{{r}_{i}}}}{{500\;{\text{AU}}{\kern 1pt} }}} \right)}^{{ - 4}}}\;\mu {{{\text{m}}}^{{ - {\text{2}}}}}\;{\text{yea}}{{{\text{r}}}^{{ - {\text{1}}}}}{\kern 1pt} . \\ \end{gathered} $$

(A.11)