Student Name: Lab TA Name:

A1101, Lab 2: Distance, Size, and Motion Laboratory Worksheet

Part 1: Angular size estimation

On the wall you will find circles that are 3 cm, 6 cm, and 30cm in diameter, and on the floor you will find tape marks at distances of 1, 2, 3, 4, and 5m.

At what distance can you just barely cover up the 3cm circle with your index finger tip held at arm’s length? Give an answer in cm, recalling that 1m = 100 cm.

At what distance can you cover up the 6cm circle with your index finger tip?

At what distance can you cover up the 30cm circle with your fist held at arm’s length?

Compute the angular size of your fingertip at arm’s length and your fist at arm’s length using the formula θ = (L/d) × 57.3 degrees where (in this case) L is the diameter of the circle and d is the distance you need to stand away to cover it. You should give two separate estimates for your fingertip (from the small and medium circles) and one for your fist.

Fingertip 1:

Fingertip 2:

Fist:

Next to your estimates for your own fingertip/fist, list the estimates that other members of your group made for their fingertips/fists.

(It should be noted that the formula used here is an approximate one, accurate only for angles that are much smaller than 57.3 degrees. In astronomy, where we are looking at very distant objects, this “small-angle” approximation is almost always a good one, but you need to be cautious in applying it to, say, architecture or engineering, where you might be dealing with larger angles. The general formula, which works for small and large angles, is (L/d) = sin(θ / 57.3 degrees) and if you’re being careful you should use 180/π = 57.29577951… instead of 57.3.)

Part 2: Angular and physical sizes

For nine different solar system bodies, the table below lists dearth, which is the closest that body comes to the Earth (in km), θbody, which is the angular diameter the body has when it is at this minimum distance (in degrees), and dsun, the distance of the body from the Sun (in km). The table is ordered in increasing distance from Earth.

Body dearth θbody dsun Diameter

Moon 3.8×105 km 0.50 deg 1.5×108 km Venus 4.2×107 km 0.0165 deg 1.1×108 km 7 8 Mars 7.8×10 km 0.0050 deg 2.3×10 km 6800 km Mercury 9.2×107 km 0.0028 deg 5.9×107 km 4500 km Jupiter 6.3×108 km 0.0130 deg 7.8×108 km Saturn 1.3×109 km 0.0053 deg 1.4×109 km 120,200 km Uranus 2.7×109 km 0.0011 deg 2.9×109 km 51,800 km Neptune 4.4×109 km 0.00064 deg 4.5×109 km Pluto 5.8×109 km 0.000022 deg 5.9×109 km 2200 km

For each of these bodies, compute the diameter, in km, using the formula Diameter = d × (θbody / 57.3 deg)

(Which distance d should you use here, Earth or Sun?)

First do the calculation for Mars and check that you get the result given in the Diameter column of the table. Then do the calculation and fill in the Diameter column for the Moon, Venus, Jupiter, and Neptune.

Write your name on the graph handout, which you will turn in together with this worksheet.

On the first graph, make a plot of each body’s angular diameter vs. its distance from the Earth. Five bodies have been plotted already; check one or two examples so that you understand why they appear where they do.

Now make the same plot on the second graph, which uses logarithmic axis markings. We will often use logarithmic axis graphs in these labs, and comparing these two examples may help you understand why.

On the third graph, make a plot of each body’s physical diameter (the Diameter column in the table) vs. its distance from the Sun (instead of the Earth).

Part 2, continued

Discuss these plots with others in your group and write (in the space below) comments on things you notice about them.

Part 3: Period vs. distance

Body dearth dsun Period (helio) Period (geo)

Venus 4.2×107 km 1.1×108 km 0.62 yr 1.60 yr 7 8 Mars 7.8×10 km 2.3×10 km 1.88 yr 2.40 yr Mercury 9.2×107 km 5.9×107 km 0.24 yr 0.32 yr Jupiter 6.3×108 km 7.8×108 km 11.9 yr 1.09 yr Saturn 1.3×109 km 1.4×109 km 29.5 yr 1.04 yr Uranus 2.7×109 km 2.9×109 km 84.0 yr 1.01 yr Neptune 4.4×109 km 4.5×109 km 164.8 yr 1.01 yr

The table above lists the same Earth and Sun distances as before, now just for the major planets. The fourth column lists the orbital period of the planet around the sun (labeled helio for heliocentric). The fifth column lists the orbital period of the planet “around the Earth” (labeled geo for geocentric); specifically, this is the time between successive conjunctions of the planet when it is in line with Earth and Sun (called the “synodic” period).

In the fourth and fifth graph of the graph set, make plots of • The “geocentric” period vs. distance from Earth • The “heliocentric” period vs. distance from the Sun Mars and Uranus are already plotted for you.

Discuss these plots with others in your group and write (in the space below) comments on things you notice about them.