Know that the force that keeps an object in orbit around the Sun is the gravitational attraction of the Sun

6.1.2 The Solar System

Learning Objective

Understand that the force which keeps an object in orbit around the Sun is the gravitational attraction of the Sun.

Key Concepts

• Gravitational force between two masses.
• Newton’s law of universal gravitation.
• Centripetal force required for circular motion.
• Balance of gravitational and centripetal forces for an orbiting body.

Newton’s Law of Universal Gravitation

The attractive force between two masses \$M\$ and \$m\$ separated by a distance \$r\$ is given by

\$F_{\text{gravity}} = G\frac{M m}{r^{2}}\$

where \$G = 6.67\times10^{-11}\,\text{N m}^{2}\,\text{kg}^{-2}\$ is the universal gravitational constant.

Centripetal Force for Circular Motion

An object of mass \$m\$ moving in a circular orbit of radius \$r\$ with speed \$v\$ requires a centripetal force

\$F_{\text{centripetal}} = \frac{m v^{2}}{r}\$

Deriving the Orbital Condition

For a stable orbit the gravitational force provides exactly the required centripetal force:

\$G\frac{M_{\odot} m}{r^{2}} = \frac{m v^{2}}{r}\$

Canceling \$m\$ and rearranging gives the orbital speed

\$v = \sqrt{\frac{G M_{\odot}}{r}}\$

or, using the orbital period \$T\$ (where \$v = \frac{2\pi r}{T}\$), we obtain Kepler’s third law:

\$T^{2} = \frac{4\pi^{2}}{G M_{\odot}}\,r^{3}\$

Typical \cdot alues in the Solar System

Why Gravity Keeps Planets in Orbit

1. The Sun’s mass \$M_{\odot}\$ is enormously larger than any planet’s mass, so the Sun exerts a strong gravitational pull.
2. Each planet moves forward due to its inertia; without a force it would travel in a straight line.
3. The Sun’s gravity continuously pulls the planet toward it, constantly changing the direction of the planet’s motion.
4. This continuous change of direction is exactly the centripetal acceleration needed for circular (or elliptical) motion, so the planet remains in orbit.

Common Misconceptions

• “Gravity only pulls objects down to the ground.” – In space, “down” is toward the centre of mass, which for planets is the Sun.
• “An object needs a continuous push to stay in orbit.” – No push is needed; the gravitational attraction provides the necessary centripetal force.

Check Your Understanding

1. State the formula that relates the gravitational force between the Sun and a planet to their masses and separation.
2. Explain in one sentence why a planet does not fall into the Sun despite the gravitational attraction.
3. Calculate the orbital speed of Earth using \$r = 1.496\times10^{11}\,\text{m}\$ and \$M_{\odot}=1.989\times10^{30}\,\text{kg}\$. (Show the steps.)

Summary

The gravitational attraction of the Sun provides the centripetal force required for planets and other objects to remain in orbit. By equating the gravitational force to the centripetal force, we obtain the relationships that describe orbital speed and period, which are fundamental to understanding the dynamics of the Solar System.