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Why does a planet's orbit have the Sun at one focus of an ellipse according to Kepler's first law?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

Kepler’s first law states that planets move in elliptical orbits with the Sun at one focus. While Kepler derived this empirically from Tycho Brahe's observations, the physical reason lies in the nature of gravitational force.123

In the context of celestial mechanics, Isaac Newton demonstrated that for any body moving under a central force that varies inversely with the square of the distance ( $1 / r^{2}$ ), the resulting path is a conic section. When the total energy of the orbiting body is negative—meaning it is gravitationally bound to the central mass—the path is mathematically constrained to be an ellipse.134

Because the Sun acts as the central point of this gravitational attraction, it occupies one of the two foci of the elliptical path. If the body's energy were higher, the path could instead become a parabola or hyperbola, which are open curves rather than closed orbits.354