With Mercury’s orbit now fully established, we can now determine the characteristics of its orbit,

such as its semi-major axis distance in physical units (AU and kilometers), it’s eccentricity, and its

orbital period in years and days. Please answer the following questions regarding the orbit of

Mercury you have drawn:

1) Is Mercury’s orbit circular? Describe its shape. How does that shape compare to a perfect

circle? [3pts]

2) Measure the length of Mercury’s major-axis and minor-axis in millimeters (mm). [4pts]

The length of Mercury’s major axis is: **___________mm. The length of Mercury’s minor axis is: _________________mm. Half of the distance of the major and minor axes define the length of the semi-major axis (a) and the semi-minor axis (b). See Fig. 1. The length of Mercury’s semi-major axis is: a = _________________mm. The length of Mercury’s semi-minor axis is: b = _________________mm. 3) Determine the semi-major axis of Mercury in physics units [5 pts] Measure the distance between the center of the Sun and Earth’s orbit in millimeters. The Earth’s semi-major axis in millimeters is: ________________mm This value corresponds to a physical distance of 1 Astronomical Unit (AU) equal to 1.496 x 108 km. What is the conversion ratio to translate Earth’s semi-major axis to AU? Conversion Ratio: Use this conversion ratio to determine Mercury’s semi-major axis distance in AU. What is Mercury’s semi-major axis distance in AU? a = ___________________AU. What is Mercury’s semi-major axis distance in km? a = ___________________km. 4) Determine the orbital period of Mercury [4 pts]. Use Kepler’s Third Law of Planetary motion and your estimate of Mercury’s semi-major axis distance to calculate the orbital period of Mercury, first in years, and then convert that to days (1 yr = 365.24 days). Please give your answer in BOTH years and number of days. 5) Determine the eccentricity of Mercury [3 pts]. Use the equation for eccentricity given in Section 1.2 to calculate the eccentricity of Mercury’s orbit. You can use any measurement of semi-major and semi-minor axes that you have determined, but be sure to make sure the units match. The eccentricity of Mercury is, e = ______________**.

6) Determine the perihelion and aphelion distances [4 pts].

Use the equations for perihelion and aphelion distance given in Section 1.2. Please

calculate these distances in AU and kilometers.

Mercury’s aphelion distance is: aphelion distance =

=

perihelion distance =

=

7) Determine the synodic period of Mercury using two different methods [4 pts].

Method 1: Use your “Orbit of Mercury Spreadsheet” to determine how days pass from

each greatest Eastern elongation to the next. You should have 7 values. Average these 7

values together to determine an estimate of Mercury’s synodic value.

Write 7 independent estimates HERE:

What is the average of those 7 estimates? Your estimate of Mercury’s synodic period is…

Recall by definition, the synodic period of an object is the time it takes for it to return to the same

geometric configuration (e.g., one greatest Eastern elongation to the next), so each one of the

seven values is an independent estimation of the synodic period. While each independent

estimate may have large error (be far from the true value), an average of many measurements

will quickly approach the actual value of Mercury’s synodic period.

Method 2 : Use the equation in Section 1.4 that relates orbital period to synodic period to

determine Mercury’s synodic period, S, using your determination of Mercury’s orbital period from

Question 4. Note this equation requires the periods to be in years. You will have to convert to

days.

Mercury’s synodic period is, S =

8) Error Analysis [13 pts].

Ask your instructor for the real value for Mercury’s semi -major axis, eccentricity, orbital

period, and synodic period. For the synodic period, record both of your estimates. Record

those values here:

Real Value Your Estimate Percent Error

Semi-major axis

Eccentricity

Orbital Period

Synodic Period 1) 1)

2) 2)

Calculate your percent error for each of these values using the equation for percent error.

% =; − ;×100

How do your values compare to the real values? Are they close to the real values?

Consider your two estimates of the synodic period. Which one is closer to the real value?

Which one do you trust more? To help frame your response, briefly describe the methods used

to determine both estimates? Which one involved more steps, each of which has its own error

associate with it?

What do you think caused the differences between your estimates and the real values?

How might you improve your estimates? That is, what steps could you take in this lab to

make your estimates be closer to the real values?

Sample Solution

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