### Assignments:

Unfinished Assignment Study Questions for Lesson 15

### Lesson Objectives:

- Newton's Universal Law of Gravitation
- Newton's Extension of Kepler's Laws
- Orbital energy
- How gravity affects tides

In astronomy, the most important force is gravity, as it is responsible for basically all large-scale motion in the universe.

Isaac Newton described how gravity works with his universal law of gravitation.

This law can be summarized in 3 statements.

First, every mass attracts every other mass through a force called gravity.

Second, the strength of the gravitational force attracting any two objects is directly proportional to the product of their masses. For example, if you double the mass of one object, you double the force of gravity between the two objects.

Third, the strength of the gravity between two objects decreases with the square of the distance between their centers. Thus, the gravitational force follows an inverse square law, meaning if you double the distance between two objects, the force of the gravity weakens by a factor of four (or two squared).

The statements we used to summarize the universal law of gravitation can be combined mathematically into the following equation. It may look complicated, but it simply states that to calculate the force of gravitational attraction, 'F', you need to multiply the mass of the two objects, divide that by the distance squared, and then multiply by what is called the gravitational constant.

As you can see in the equation, since the distance is squared, if you double the distance, the force is weakened by a factor of four. If you triple the distance, it would be weakened by a factor of nine. This demonstrates why the universal law of gravitation is an inverse square law.

By the time Newton published his laws in 1687, Kepler's laws of planetary motion had been accepted for over 70 years. Newton's findings extended Kepler's laws.

First of all, Newton discovered that Kepler's first two laws applied to ALL orbiting objects, not just planets orbiting our Sun. This includes moons around planets, satellites around the Earth, and asteroids around the Sun.

Secondly, he discovered that an ellipse is NOT the only possible orbital path. Ellipses are the only path for *bound* orbits, which refer to orbits in which the object goes around the other object over and over again - in other words, the two objects are bound together.

But for unbound orbits, which is where an object comes close to another object just once, the orbit will take a hyperbolic or parabolic path, such as when a comet enters our solar system and loops around the Sun once, never to return.

In addition to his work on Kepler's first two laws, Newton created a more generalized version of Kepler's third law. His version allows us to calculate the mass of distant objects if we can measure the orbital period and the distance between the two objects. Newton's version of Kepler's third law is the primary way that we determine masses throughout the universe.

Orbital energy is the sum of the kinetic energy and gravitational potential energy of an orbiting object. Due to the law of conservation of energy, the amount of orbital energy always stays the same unless something causes it to gain or lose orbital energy. It cannot change spontaneously and an object's orbit can only change if it gains or loses orbital energy.

Gravitational encounter is an example of how orbital energy can be changed. Gravitational encounter occurs when objects pass near enough to each other to feel the effects of each other's gravity. For example, if a comet is heading for an unbound orbit of the Sun but passes close to Jupiter, it would exchange energy with the planet. The comet would lose energy while Jupiter would gain energy. If the comet loses enough energy and gets pulled in close enough, it may change from an unbound orbit of the Sun to a bound and elliptical orbit. However, the effect on Jupiter would be practically nonexistent because of the difference in mass.

Another way orbital energy is changed is through atmospheric friction. For example, when a satellite is orbiting the Earth, it loses orbital energy due to friction with the particles in the Earth's thin upper atmosphere. Eventually, if it loses enough orbital energy, it will plummet towards the Earth and crash.

On the other hand, a spacecraft can escape the Earth's orbit by increasing its orbital energy. In that case, we use a rocket to increase its orbital energy until it has enough to escape Earth's orbit. The energy required to do this is called escape velocity.

The escape velocity is about 40,000 km/hr if a spacecraft starts near the surface. This escape velocity does not depend upon mass but is affected by where you start (at the surface or higher above). Gravity weakens with distance so it would take less energy to escape from high in the Earth's atmosphere than from the Earth's surface.

How does gravity affect tides? Tides rise and fall because gravity attracts the Earth and Moon towards each other.

The difference in attraction as we go from the side of the Earth facing the Moon to the side facing away causes a tidal force.

Tidal forces stretch along the Earth-Moon line, creating two tidal bulges. The two daily high tides occur as a location on Earth rotates through the two tidal bulges, while the points halfway between the two tidal bulges mark the location of the low tides.

The Sun also affects the tides but given its greater distance away from the Earth, its effect is less prominent. When the tidal forces from the Sun and Moon work together, we get larger *spring* tides, and when they work against each other, we get smaller *neap* tides.