Purpose: To familiarize students with the various components that determine a planet’s movement around the Sun. Goals

Name _________________________ Date: ________
Planetary Properties - Orbits
Purpose: To familiarize students with the various components that determine a planet’s movement around the Sun.
Goals: Upon successful completion of this lab assignment, students will be able to:

1. 
   Use Newton’s Law of Gravity to calculate the forces between the Sun and the planets.
   
2. 
   Understand the physical significance of planetary properties.
   
3. 
   Understand the dependency between certain planetary properties.
   
4. 
   
   

Newton’s Law of Gravity
Isaac Newton (1642 - 1727) first proposed the idea that objects fall to the ground because of the force of gravity. Gravity, Newton says, is an inherent property of any massive object. As long as an object has a mass, it will pull on other objects with mass. Gravity, like all other forces, is a vector quantity in that we need to know the magnitude (i.e. amount) of the force as well as the force’s direction. Luckily, the direction of the gravitational force is always the same: toward the center of the objects in question. The magnitude of the force can be calculated by using the Universal Law of Gravitation:



Example:
What is the magnitude of the force of gravity between the Earth and the Sun?
To calculate the force of gravity between the Sun and Earth we will need to know the following information, which is available in a textbook or on an appropriate website:

To calculate the magnitude of the force of gravity, simply substitute the values in the appropriate places in the Universal Law of Gravitation and then follow through with the calculation.

According to Newton’s 3^rd Law of Motion, the force of 3.52 x 10²² N is felt equally by both the Sun and the Earth, but in opposite directions. The Sun is being pulled toward the center of the Earth with 3.52 x 10²² N while the Earth is being pulled toward the center of the Sun with the same force. How each object reacts to the force is different and can be understood by looking at the Law of Inertia. Since the Sun is much more massive than the Earth, the Sun will resist this force more intently than the Earth. As such, it is adequate to visualize the Earth being held by the force of gravity in a closed orbit around a nearly motionless Sun.

1. 
   Use the Universal Law of Gravitation to calculate the force of gravity between the Sun and Jupiter. Show the steps in your calculation; report your answer in scientific notation.
   

2. 
   Use a ratio to calculate how many times greater the force of gravity between the
   

Sun and Jupiter is than the force of gravity between the Sun and Earth. Show the

steps in your calculation; report your answer in scientific notation.

According to Newton’s 2^nd Law of Motion (the Law of Inertia), a planet that is moving at some velocity (v) will do so in a straight path so long as an outside force doesn’t act on it. However, planets are under the influence of the force of gravity brought on by the presence of the Sun’s mass. This centralized force causes the planet to be pulled (i.e. accelerated) toward the Sun’s center (Figure 7.3). This combination of the planet’s desire to wander a straight path through space with its experiencing centripetal acceleration, the planet ends up being trapped in a closed orbit around the Sun.

Orbital Properties
Planets move around the Sun in closed paths, referred to as orbits. Certain properties of a planet’s orbit can affect the probability of whether or not life will develop. The properties of orbital motion have been well understood ever since Johannes Kepler (1571 - 1630) first proposed his three laws of planetary motion nearly four centuries ago. In this section we will investigate each of Kepler’s laws of planetary motion in order to better understand how spacecraft are sent to other worlds in our solar system.
Kepler’s 1^st Law: Shapes of Orbits
Kepler’s first law identifies the shape of the orbital path of a planet as it travels around the Sun is that of an ellipse and not a circle. An ellipse differs from a circle in that it has two focal points (foci) that are both the same distance from the center, whereas a circle has only one focal point located on the exact center (see Figure 1.2). This means that for an ellipse, there are axes of many different sizes that can be drawn through the center. There are two important axes that are important references for an ellipse. The minor axis is the shortest axis that can be drawn across the ellipse through the center while the major axis is the longest.

Kepler also noted that the Sun is located at one of the focal points of a planet’s elliptical orbit. This means that the distance between the Sun and a planet is always changing as the planet moves around its orbit. As a standard way of comparing orbit sizes, the average Sun-planet distance is always referenced. Below are several important properties to know when discussing planetary orbits:

Semi-major axis (a): half the length of the major axis; this distance is equivalent to

the average Sun-planet distance.

Eccentricity (e): defines the “shape” of the orbit by taking the ratio of the Sun

center distance (c) to the semi-major axis (a){e = c/a}. See Figure 7.5.

Periapsis: the point in an orbit that is closest to the body being orbited.
Apoapsis: the point in an orbit that is farthest from the body being orbited.
Periapsis and apoapsis are general terms used when discussing orbits around an unspecified body. More specific terms are used when the body being orbited is known. For example, the terms perihelion and aphelion are used to refer to the point in a planet’s orbit that is closest and furthest from the Sun, respectively.


Kepler’s 2^nd Law: Equal Area in Equal Time
Kepler’s Second Law of Planetary Motion states that an imaginary line connecting the Sun and planet will “sweep out” equal areas within the orbital plane within equal units of time. Though this is strictly an observation of planetary behavior, looking at this discovery in terms of Newton’s laws reveals a deeper significance.

As a planet moves along its elliptical orbit, the distance separating the planet and the Sun grows and shrinks over time. This change in the separation means that the force of gravity between the two also changes. The result is that a planet will end up moving faster in its orbit when it is closer to the Sun and slower when it’s further away.

Kepler’s 3^rd Law: P² = a³
Kepler’s Third Law of Planetary Motion draws a direct correlation between the orbital period of a planet (measured in years) to its average distance from the Sun (measured in AU). Though the accuracy of this law is improved upon using Newton’s laws, Kepler’s third law in its stated form will be useful enough for our investigations.
Example: The orbital period of Venus is measured to be 0.6152 years. What is the average distance between the Sun and Venus?
Solution:

Use the orbit of a fictitious planet orbiting a generic star in Diagram 1.1 to answer Questions 3 – 11. Show the steps in each calculation; report your answer in scientific notation where appropriate.

3. 
   Draw in the major axis of the planet’s orbit.
   

4. 
   Mark the center of the major axis with an ‘x’.
   

5. 
   Use a ruler to determine the length of the semi-major axis and the Sun-center
   

distance.

Semi-major axis (a) = ________ cm Sun-center distance (c) = ________ cm

6. 
   Calculate the eccentricity (e) of the orbit.
   

e = ________

7. 
   Label the point of perihelion and aphelion on the planet’s orbit.
   

8. 
   If the scale of Diagram 1.1 is, 1 cm = 3 AU, determine the planet’s true average
   

distance from the star it is orbiting.

True average distance = ________ AU

9. 
   Between which two planets would this planet be located if it were orbiting within
   

our solar system?

10. 
    Calculate the orbital period for the planet.
    

P_orb = ________ yrs

11. 
    Where will the planet be when it is moving with its greatest velocity?