Mercury’s Orbit

Name:____________________________  Lab Section: Online  Date: _____________

Kepler’s First Law

• Circle any three points on the orbit plot that are widely spaced out and label them P1, P2 and P3.  Use a ruler to draw a line from F1 to P1 and then draw another line from P1 to F2.    Measure the exact length of each line segment in centimeters.  Add the two lengths to find the total distance.  Repeat this procedure for points P2 and P3.  Record your measurements and sums in the table below. 

Point	Distance (cm) F1 to Point	Distance (cm) Point to F2	Total Distance
P1
P2
P3

• Now measure the length of the major axis in centimeters. _________________
• What can you conclude from these measurements?
• The Eccentricity of an ellipse is a measure of its "flatness".  Eccentricity is defined as the ratio of the distance between focal points to the length of the major axis.

The numerical range of eccentricity can have values between 0 and 1.  An orbit with an eccentricity of 0 is a perfect circle.  A value of say, 0.9, is a very flat ellipse which is typical of a comet’s orbit.

• Determine the eccentricity of Mercury’s orbit below.

• How would you describe Mercury’s orbit (circle best description)?

Low Eccentricity (close to a circle) / Moderate Eccentricity / High Eccentricity

Kepler’s Second Law

Time from A to B = Time from C to D Area 1 = Area 2

The point at which a planet is closest to the Sun is called perihelion and the greatest distance is called aphelion.  Both points are located on the major axis.  Circle and label the perihelion and aphelion points for Mercury’s orbit.

• At what point in its orbit is the planet Mercury moving fastest?

Explain why that point was chosen.

In the following exercise Kepler's 2nd law will be verified by the method of equal areas.  Circle a point on the orbit above the major axis but close to perihelion and label it “A”.  Now count 5 time intervals from point “A” in a counterclockwise direction.  Circle that point and label it “B”.  Imagine Mercury moving in a counterclockwise direction from “A” to “B”.  As the planet orbits, a line from the planet to the sun will sweep out an area.  The planet moving through the same time interval anywhere on the orbit will sweep out the exact same area.  This is the essence of Kepler’s second law.

Circle another point below the major axis but this time near aphelion.  Label that point “C”.  Count another 5 time intervals from point “C” again in a counterclockwise direction.  Circle that point and label it “D”.  Repeat this process for the third time using the region below the major axis and about halfway in between aphelion and perihelion.  Label those points “E” and “F”.  To find the approximate area for each time interval use the area formula for a triangle.

Area = ½ (base x height).

To form a triangle draw a line from the Sun’s center to point “A” and another line from the Sun’s center to point “B”.  Connect points “A” and “B” to complete the first triangle and label it “Area 1”.  Repeat the procedure for the other two areas and label them “Area 2” and “Area 3”.  The base distance is the longest side of the triangle.  The height of the triangle is a perpendicular line from the base to the vertex opposite the base.  Measure the length of the base and the height for each triangle and record those values in the table below. The final step is to calculate the area using the formula above. 

Area 1                        Base = ___________ Height = ___________ Area = ___________

Area 2                        Base = ___________ Height = ___________ Area = ___________

Area 3                        Base = ___________ Height = ___________ Area = ___________

Do these measurements support Kepler’s 2^nd Law?  Explain your results.

Kepler’s Third Law

Complete the following exercise.  Show your work and use the proper units and significant figures.

• Measure the semi-major axis of Mercury’s orbit in AUs using the AU scale provided.  Record the value below.

• Using Kepler’s Third Law calculate the orbital period of Mercury in years.

Solve for P Þ

• Find the orbital period in days (multiply the period in years by 365 days/year).

P_days =

Find the orbital period in hours (multiply the period in days by 24 hours/day).

P_hours =

Now let’s “weigh” the sun!  For any two bodies orbiting each other Newton generalized Kepler’s third law to read:

,

where G is the gravitational constant,  M the Sun’s mass and m the planet’s mass.

This form of Kepler’s Third Law is frequently used to find the total mass of any two objects orbiting around each other.  For example, Mercury’s orbit can be used to find the Sun’s mass.  Solving the above equation for M (mass of Sun) gives:

,

Since the mass of a planet is insignificant compared to the Sun’s mass, m + M was set equal to M.

• Using the values that were determined previously for Mercury’s semi-major axis (a) and orbital period (P), calculate the Sun’s mass by completing the following steps (show your work for each step and don’t forget the units).

To use the third law first convert the period in years to seconds.  Multiply the period in years by 3.16 x 10⁷ seconds/year.

P =

Now square the period P.

P² =

Measure the length of the semi-major axis using the kilometer scale provided and record the value below.

Convert the semi-major axis from kilometers to meters by multiplying the
semi-major axis by 1000 meters/km.

a =

Cube the semi-major axis.

a³ =

Now that you have both P² and a³ in the proper units solve for the Sun’s mass using Kepler’s Third Law.  The numerical value of is .

Kepler’s Third Law: