The Hill Radius

The Hill radius is a component of the sphere region which determines the orbit of a test particle of relative minimal mass around a planetesimal.  The Solar Nebular Disk Model predicts nebular disks, with small grains of dust and ice particles, to develop larger objects of appreciable size.  These objects begin to gravitationally influence other material in the disk.  In order for a planetesimal to retain an object in its orbit the object must pass within the planetesimal's Hill sphere.  The Hill Sphere as well as the Nebular Disk Model helps with the understanding of the formation of planetary and stellar systems.

Let's begin by assuming a circular orbit of a test particle around a planetesimal of relative mass much greater than the mass of the test particle in question (m << M).  From Kepler's Third Law the orbital period of the test particle is given by:  

We define the Hill radius as the distance from where the orbital period of the test particle around the planetesimal is equal to the orbital period of the planetesimal around the Sun.  Suppose at a distance d, the orbital period of the planetesimal around the sun is equal to the orbital period of the test particle around the planetesimal.

Setting their orbital periods equal gives:

Doing a bit of algebra we then have the Hill Radius as:

Rewritten with term alpha,

Where term alpha is defined by the planetesimal and sun's mass-densities:

In applications, the Hill Sphere describes the gravitational sphere of influence of a planetesimal in a multiple body system such as the two-body system of the Sun and Earth.  Supposing a particle comes within the vicinity of a two-body system with one larger and one smaller object in relative masses.  If the particle passes through the Hill sphere of the smaller object with a relatively low velocity, the particle will then be tied to the smaller not larger object's orbit.  The continual collection of particles by an object will cause the object's radius to grow as well as its Hill radius.  

Calculating Earth's Hill Radius

The Earth and Sun have mass densities:

                                 

The radius of the Sun and Earth's distance to the sun:

                              

This means if an object came within .01 AU of the Earth at a relatively low velocity it would become caught within Earth's orbit and not the Sun's orbit.

In this diagram the Hill Sphere's are denoted by the
circular regions surrounding the Earth and Sun.

Sources: Wikipedia: Nebular Theory, Hill Sphere, Orbital Period

Carroll: The Hill Radius