Satellite orbits lie in planes that bisect the orbited body. If the Earth were not rotating, each orbiting satellite would pass over the same point on the Earth with each orbit, crossing the equator repeatedly at the same longitude.

Because the Earth is constantly rotating, each orbital pass of the satellite (as indicated by the model described in this activity) appears to be to the west of the previous one. In reality, the Earth is rotating eastward as the orbital plane remains fixed.

### Students will examine the factors determining the length of a satellite's orbit around Earth

Recognize that the Earth rotates 360 degrees in 24 hours, or:

60 minutes X 24 hours = 1440 minutes --------- -------- ------------ 1 hour 1 day 1 day

Dividing 360 degrees by 1440 minutes shows that the Earth is rotating 0.25 degrees every minute. Here's the math:

360 degrees = .25 degrees ----------- ----------- 1440 minutes 1 minute

The satellites that we will want to track travel around the Earth in approximately 102 minutes. Thus, we can see that if the satellite crossed the equator at 0 degrees longitude on one orbit, it would cross over 25.5 degrees longitude 102 minutes later.

25 degrees = 25.5 degrees ---------- ------------ 1 minute 102 minutes

The extremely large size of the Earth in relation to the very modest thickness of the atmosphere leads to frequent, intentional distortions of scale in map projections. Constructing a true scale physical model of an orbiting satellite's path will lay the groundwork for insights into geographical configurations on a three-dimensional sphere, and the physical characteristics of the satellite's orbit.

Given the following information answer the questions that follow.

If you are tracking the NOAA series of weather satellites, the following figures will be a close approximation.

Km. Miles Mean orbital altitude 860 534 Width of field of view 2900 1800 Orbital period...102 minutes

Determine the scale of your globe by measuring either its diameter or circumference and comparing that to the Earth's actual diameter or circumference.

What follows is a series of questions to test for understanding. The answers to each question follow on an accompanying page.

### Questions

- What is the Earth's diameter? What is the diameter of the globe?
- What is the ratio of the model diameter compared to the Earth's diameter? This is your scale measure.

Using a piece of wire (#10 works well), position the wire in such a way as to center the wire over your location on the globe. For our location here in Maine, the wire should cross the equator at approximately 60 degrees west longitude, and continue up to the left of the north pole by about 8 degrees. Experiment with different ways of supporting the wire slightly above the globe. With our globe, we were able to rig a support from the globe support bar already in place. The height of the globe support bar was almost at the exact height as our orbital plane, which we will be discussing shortly. The globe should be able to rotate under the wire. You may want to add a piece of plastic transparency material to the wire. This will represent the width of the Earth that the satellite will image on a typical pass. Because this width is approximately 1800 miles, the scale plastic strip will be approximately 2.5" wide.

- Just how high should we position the piece of wire above the globe?

On a large sheet of paper, draw a circle of the same diameter as your globe.

This circle will represent the surface of the Earth. Write the scale of measure in the lower right-hand corner as a legend. Draw a circle having the same center as the first, but with a radius of 534 miles more than your first circle. This will represent the orbit of the satellite over the Earth's surface. Label point H on the inner circle (Earth's surface) as your town or city. Draw a straight line through this point. That will represent the horizon as it appears from your location.

The satellite we wish to examine can only be received while in an unobstructed straight line from the antenna; thus, it can only be received while above the horizon. The point at which we first receive a satellite's signal is known as Acquisition of Signal (AOS), and the point at which we lose the signal is referred to as Loss of Signal (LOS). A good analogy would be to think of sunrise as AOS and sunset as LOS.

On your drawing, label two points that lie directly under the points at which the satellite will come into (Aquisition of Signal, AOS) and go out of (Loss of Signal, LOS) receiving range.

Refer to your diagram and answer the following questions:

- How many miles from your location are the points on Earth over which the satellite will come into or leave receiving range?

You may wish to draw a circle on your globe to represent this range, known as the acquisition circle.

- Knowing the period of a complete orbit, find a way to calculate the amount of time that the satellite will be in range if it passes directly overhead as you've illustrated.

Rotate the globe to position the wire so that the northbound orbit will cross the equator at 0 degrees longitude. If the Earth were not rotating, the satellite would always follow the path illustrated by the wire. Would this path ever bring the satellite over your school?

Polar orbiting weather satellites have an orbital period of about 102 minutes. This means they complete a trip around the world in approximately 1 hour and 42 minutes.

The Earth rotates 360 degrees in 24 hours.

- How many degrees does the Earth rotate in 1 minute?
- How many degrees does the Earth rotate in 1 hour?
- How many degrees does the Earth rotate in 102 minutes?
- If the satellite crossed the equator at 0 degrees longitude at 0000 UTC, at what longitude would it cross the equator 102 minutes later? 204 minutes later?
- How many orbits will the satellite complete in one day?
- How many miles does the satellite travel during one orbit?
- How many miles does the satellite travel during each day?
- How many times will your location be viewed by this satellite in one day?

### Answers

#### Question 1

The sample calculations given here are based on the use of a 12-inch diameter globe and should be proportionally adjusted for use of other materials. The diameter of a globe can be determined by first finding the circumference using a tape measure.

Earth Model Diameter 8100 miles 12 inches Circumference 25,500 miles 37.75 inches Path width 1800 miles 2.5 inches Orbit altitude 534 miles .75 inches

#### Question 2

Using the circumference, a ratio can be established as follows:

37.75 inches = 1 inch ------------- ------- 25,500 miles 675 miles

#### Question 3

By the scale established above, the orbital height of 500 miles is represented by a scale distance of:

1 inch = .75 inch ------- -------- 675 miles 500 miles

Thus, on this model of the Earth, using a 12-inch diameter globe, the wire representing the orbital plane should be placed approximately 3/4" above the surface of the globe.

#### Question 4

Assume the satellite pass is directly overhead of point H (your home location). R equals the radius of our reception area (acquisition circle) and D equals the diameter of the reception area.

Having established the AOS and LOS points on the orbital curve, project lines from both the AOS and LOS points down to the center of the Earth. Label this point G (Earth's Geocenter). With these lines in place, go back and label the two points where these two lines intersect the Earth's surface. Appropriately label these Points A and L. If we measure the angle formed by points AGL, we find it to be 55 degrees. This is angle D (diameter of acquisition circle). Angle R is half of angle D (R = radius of acquisition circle). Why is it important to know this angle? We are interested in knowing the size of our acquisition circle, that is, the distance from our home location (point H) that we can expect to receive the satellite signal.

We've previously determined that the circumference of the Earth is approximately 25,500 miles. Thus:

360 degrees X 55 degrees ----------- ---------- 25,500 miles 3800 miles

We know that the satellite signal will be present for 55/360 of this distance. Using your calculator, 55/360 represents .1528 of the total circle. Thus, if we multiply the Earth's circumference (25,500 miles) by .1528, we can determine the distance (diameter) of our acquisition circle, which in this case is equal to 3896 miles. Half of that, or R, is 1948 miles. For a receiver in Maine, a satellite following the path as indicated on Figure A would be somewhere south of Cuba when the signal is first heard (AOS), and to the north of Hudson Bay when the signal is lost (LOS).

#### Question 5

Let's look at some numbers. Remember that it takes 102 minutes for a NOAA-class satellite to make one complete orbit around the Earth. We now want to determine the fractional part of the orbit, the exact time inside our acquisition circle, that the signal will be usable to us. Using the same math as before, the satellite will be available for 55/360 of one complete orbit. As previously defined, 55/360 = .1528. This number times the orbital period of 102 minutes yields 15.6 minutes. This means that on an overhead pass, we can expect to hear the satellite's signal for approximately 15.6 minutes. Using a reliable receiver with outside antenna will in fact yield the above reception time.

#### Question 6

The Earth rotates 360 degrees in 1440 minutes. Thus:

360 / 1440 = .25 degrees/minute

#### Question 7

If the Earth rotates .25 degrees each minute, then:

.25 x 60 = 15 degrees/ hour

#### Question 8

If the Earth rotates .25 degrees each minute, then:

.25 x 102 = 25.5 degrees/orbit

#### Question 9

The satellite crosses the equator at 0 degrees west longitude. The next equator crossing will be 102 minutes later, and located 25.5 degrees west. Thus, the next equator crossing will take place at 25.5 degrees west longitude. For the next orbit, it will again be 25.5 degrees further west, which would place this crossing at 51 degrees west longitude (25.5 + 25.5).

#### Question 10

The satellite will be on its 15th orbit at the end of 24 hours since it completes 14 orbits in one day and begins a 15th. More precisely:

1 orbit X 1440 minutes = 14.12 orbits ------- ------------ ------------ 102 min. 1 day 1 day

#### Question 11

The satellite will travel the equivalent of the circumference of each orbit, or 25,500 Earth miles. However, the circumference at orbital altitude is approximately 38,850 miles.

#### Question 12

25,500 miles/orbit x 14.12 = 360,060 miles/day (approx.). These are miles traveled at ground level. More precisely, the circumference of the orbital circle is about 28,850 miles, thus:

28,850 miles/orbit x 14.12 = 407,362 miles/day

#### Question 13

At least four times/day. The satellite will usually come within range on two consecutive orbits, sometimes three. Usually figure on a pass to the east of your location, nearly overhead, and then to the west. Remember that at one part of the day the satellite will be on an ascending pass (crossing the equator going north) and at another time of the day the satellite will be on a descending pass (crossing the equator going south).

Here is one final activity which will test your understanding of orbital parameters. Assume you are tracking a NOAA-class satellite with an orbital period of 102 minutes, and a longitudinal increment of 25.5 degrees west/orbit.

Complete the chart for the remaining five orbits.

Orbit # Equator Crossing (EQX) Time 1 0 degrees 1200 UTC 2 3 4 5 6

Here's what you should have:

2 25.5 degrees 1342 UTC 3 51.0 degrees 1524 UTC 4 76.5 degrees 1706 UTC 5 102 degrees 1848 UTC 6 127.5 degrees 2030 UTC

### Materials

- 12-inch globe
- length of #10 wire (20" to 40") to project orbital plane
- clear plastic strip to project width of Earth image
- data for a NOAA-class satellite
- ruler
- protractor
- compass