The perihelion and aphelion of the asteroid Apollo are 0.964 ×10^8 km and 3.473 ×10^8 km from

step 1 For this problem, we are told that the asteroid 433, or Eros, is the second largest near -Earth

The perihelion and aphelion of the asteroid Apollo are 0.964...

The perihelion and aphelion of the asteroid Apollo are $0.964 \times 10^8 \mathrm{~km}$ and $3.473 \times 10^8 \mathrm{~km}$ from the sun, respectively. Apollo therefore swings in and out through earth's orbit. Find (a) the semi-major axis, (b) the period of Apollo's orbit in years, given earth's semi-major axis $a_E=149.6 \times 10^6 \mathrm{~km}$. (Apollo is only one of many "Apollo asteroids" that cross earth's orbit. Some have struck the earth in the past, and others will strike it in the future unless we find a way to prevent it.)

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Key Concepts

Elliptical Orbits

Celestial bodies orbit in elliptical paths rather than perfect circles. An ellipse is defined by its two foci, and the orbit of a planet or asteroid around a star is generally an ellipse. Understanding this geometry is critical because it dictates how the distance from the focal point (like the sun) changes throughout the orbit, influencing gravitational forces and orbital speed.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse and serves as a key parameter in defining the size of the orbit. It represents half the longest diameter of the ellipse. In orbital mechanics, the average distance of the orbiting body from the central mass is given by the semi-major axis, making it a fundamental quantity for further calculations, such as determining the orbital period.

Orbital Period and Kepler's Third Law

The orbital period is the time taken by a celestial body to complete one full orbit around its central mass. Kepler's Third Law relates the orbital period to the size of the orbit, stating that the square of the period is proportional to the cube of the semi-major axis of the orbit. This law enables the calculation of the period when the semi-major axis is known, providing a vital link between the geometry of an orbit and its dynamic behavior.

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