Orbit Approximation

by Carl Martikean

Your Assumptions | Effect of the Central Force | Scale the Plot | Computing Dv| Making the Plot | Prepare for Discussion | More Things to Do

This experiment is an interesting variation a centripetal force experiment. In the centripetal force experiment, a mass, usually a rubber stopper is rotated on a string and the period of rotation is used to verify that there is a central force. Perhaps a plot of the orbit either of Mercury or of Mars is performed on Polar graph paper to examine the orbital eccentricity. (Kepler was lucky that he studied Mars first!) For an interesting synopsis of the historical significance of Comets see this.

Newton used this concept of a centrally directed force to analyze the motions of orbiting bodies. What I propose here is to develop an approximate orbit of a comet or other satellite. The plot can be used to verify Kepler's law of areas. Based on your knowledge of Newton's laws of motion:

Q1. What would you predict for the path of a ball rolling over a smooth level surface?

Q2. Would the path of the ball change if you were to strike it from one side?

Q3. Would the speed change? Suppose you gave the ball a series of "sideways" blows of equal force, what do you predict that the path would be?

Your Assumptions Top of Page

A planet or other satellite in orbit has a continuous force acting on it.  However as the body moves, the magnitude and the direction of this force change.  (Once again, refer to your study of centripetal force.)  To predict exactly the orbit under the application of this continually changing force requires the use of advanced mathematics.  We can achieve a reasonable approximation to the orbit by breaking the continuous attraction into many small steps.  In this method, the force acts as a sharp "blow " toward the sun every sixty days.  

The application of repeated steps is known as "iteration."  it is a powerful technique for solving problems. Modern high-speed digital computers use this operation to solve complex problems, such as the best path (paths) for a space probe to follow between the earth and say Mars, for example.  

In order to proceed with the plot, several additional assumptions need to be made.

  1. The force on the comet is an attraction toward the sun.
  2. The force of the blow varies inversely with the square of the comet's distance from the sun.
  3. The blows occur regularly at equal time intervals, in this case, 60 days.  The magnitude of each brief blow is assumed to equal to total effect of the continuous attraction of the sun throughout the 60 day interval.

Effect of the Central Force Top of Page

Newton's second law showed that the gravitational force will cause the comet to accelerate toward the sun.  If a Force (F) acts for a time interval, Dt, on a body of mass m then :

F=ma or m Dv/Dt  and therefore

Dv=( F/m)D

This equation relates the change in velocity to the body's mass, the force and the time for which it acts.  In our situation both the mass and Dt  are constant.  The change in velocity is thus proportional to the force.  But remember that the force is not constant in magnitude.  The force varies inversely with the square of the distance between the comet and the sun. 

Q4.  Is the force of a blow given to the comet when it is near the sun greater or smaller than one given when the comet is far from the sun?

Q5. Which blow causes the biggest velocity change?

In Fig. 8-2a the vector, v 0 , represents the comet's velocity a point A.  During the first 60 days the comet moves from point A to point B (Fig. 8-2b).  At B a blow causes a velocity change, Dv1, (Fig. 8-2c).  The new velocity after the blow is the sum of v0 + Dv1,and is found by completing the vector  triangle (Fig. 8-2d.  The comet then leaves point B with velocity v1 and continues to move with this velocity for another 60-day interval.  Because the time intervals between blows are always the same (60 days ), The displacement along the path is proportional to the velocity, v.  You therefore use a length proportional to the comet's velocity to represent its displacement during each time interval (Fig 8-2e).Each new velocity is found, as above, by adding to the previous velocity the Dv given by  he blow.  In this way, step by step, the comet's orbit is built up.

Scale the Plot Top of Page

Orbit shape is dependant on the initial position and velocity, and on he force acting.  let's assume that the comet is first spotted at a distance of 4AU from the sun.  Also assume that the comet's initial velocity is at this point is v= 2AU per year ( about 20,000 miles per hour) at right angles to the sun-comet distance R.

The following scale factors will reduce the orbit to a scale that fits conveniently on a 16' X 20" piece of graph paper.  (This can be made from four pieces of 8.5" x 11" graph paper.)

  1. Let 1AU be scaled to 6.5 cm (about 2.5 inches ) so that 4AU is 25 cm (about 10 inches)
  2. Since the comet is hit every 60 days, it is convenient t express the velocity in AU per 60 days.  Suppose you adopt a scale factor in which the velocity vector of 1 AU /60 days is represented by an arrow 6.5cm (about 2.5 inches) long.

The comet's initial velocity of  2AU per year can be given as 2/365 AU x 60 or 0.33AU  per  60 days.  This is the displacement of the comet in the first 60 days. This scales to an arrow 2.11cm (0.83in) long.  

Computing Dv Top of Page

On the scale and with the 60 day iteration interval, the force field of the sun is such that the Dv given by a blow when the comet is 1 AU from the sun is 1 AU/ 60 days.

To avoid computing Dv for each value of R (distance from the sun), make a plot of Dv against R on a graph. Then for any value of R you can immediately find the value for Dv.

Table 1 gives the values for R in AU, in centimeters, and in inches to fit the scale of your orbit plot. The table also shows for each R the corresponding value of Dv in AU/60 days. in centimeters, and in inches to fit the scale of your orbit plot.

Table 1 Scales for R and Dv

Distance from the sun, R Change in speed, Dv
AU/60 days centimeters inches AU/60 days centimeters inches
0.75 4.75 1.87 1.76 11.3 4.44
0.8 5.08 2.00 1.57 9.97 3.92
0.9 5.72 2.25 1.23 7.80 3.07
1.0 6.35 2.50 1.00 6.35 2.50
1.2 7.62 3.0 0.69 4.42 1.74
1.5 9.52 3.75 0.44 2.82 1.11
2.0 12.7 5.0 0.25 1.57 0.62
2.5 15.9 6.25 0.16 1.02 0.40
3.0 19.1 7.50 0.11 0.71 0.28
3.5 22.2 8.75 0.08 0.51 0.20
4.0 25.4 10.00 0.06 0.41 0.16

Graph these values on a separate sheet of graph paper at least 25 cm (10 in) long. Carefully connect the points with a smooth curve. This curve can be used as a simple graphical computer. To use the graph, cut off the bottom margin (or fold it under) along the R axis. (This is the x-axis of the graph) Lay this edge on the orbit plot and measure the distance from the sun to a blow point (point B in Fig. 8-4). Then using dividers or a drawing compass pick off the value of Dv corresponding to this R and lay off this distance along the radius line toward the sun (Fig. 8-4).

Making the Plot Top of Page

On your large graph paper:

  1. Hold the paper horizontally and mark the position of the sun about half way up and 30 cm (12 in) from the RIGHT edge. This is line SA.
  2. Locate the point where you first "see" the comet at a point 25 cm (10in or 4AU) to the right of the sun on SA.
  3. To represent the comet's initial velocity, draw vector AB perpendicular to SA. This vector should be 2.11cm (0.83 in). (See Scale of the Plot above.)
  4. B is the position of the comet at the end of the first 60 day interval. It is at B that the blow is struck which causes a change in velocity, Dv.
  5. Use the Dv graph to measure the distance from the sun at S, and to find Dv1, for this distance (Fig. 8-4).
  6. The force and thus the Dv is always directed toward the sun. From B lay off Dv1 toward S. Call the end of this short line M.
  7. Draw the line BC'. Thisis a continuation of AB and has the same length as AB.This represents the position of the comet at the end of the next 60 days if no blow had been experienced at B.
  8. The new velocity is the sum of the old velocity, vector BC', and Dv, vector BM. To find the new velocity, complete the parallelogram MBC'C, where C is the fourth vertex. (Remember that CC' and BM are opposite parallel sides as are BC" and MC.) Line BC (the diagonal) represents the new velocity vector v1. This is the velocity with which the comet leaves point B.
  9. Now comes the iteration .
  10. The comet moves with uniform velocity, v1 , for 60 days arriving at point C. Its displacement in that time is Dd1=v1x 60 days. However, because of the scale factor chosen, the displacement is represented by line BC. The new radius is SC.
  11. Repeat the previous steps to establish point D and so forth for at least 14 or 15 iterations. This is about half of an orbit.
  12. Connect points A, B, C... with a smooth curve.
  13. Your plot is finished.

Prepare for Discussion Top of Page

Since you derived the orbit od this comet, you may name the comet. Check your understanding of Kepler's Laws.

Q6. From your plot, find the perihelion distance?

Q7. Find the center of the orbit and calculate the eccentricity of the orbit?

Q8. What is the period of revolution for your comet?

Q9. How does the comet's speed (really velocity- why?) change with its distance from the sun?

Q10. Is Kepler's law of ellipses confirmed? (Is there a way to test your curve to see how nearly an ellipse it is?)

Q11. Is Kepler's law of equal areas confirmed? To answer this remember that the time interval between blows is 60 days, so the comet is at positions B, C, D..., etc., after equal time intervals. Draw a set of lines from the sun to each of these points, include A, and you have a set of triangles.

Find the area of each triangle. There are at least two ways. One is to use the formula for the area of a triangle, A=1/2 x b x h, where b is the base and h is the height of the triangle. The number of squares in each triangle is another method of calculating the areas. Here is a discussion of the Mathematics of Satellites.

More Things to Do Top of Page

  1. The graphical technique you have practices can be used for many problems. You can use it to find out what happens if different initial speeds and/or directions are used. You may use the same graph you prepared or you may construct a new graph. To do this, use a different law (e.g. force is proportional to 1/R3 or to 1/R or to R) to produce different paths. It is important to note that actual gravitational forces are not represented by such force laws.
  2. Use your original force graph but reverse the direction of the force to make it a repulsion. This will give you the oppoptunity to examine how bodies move under such a force. Do you know of the existence of any such repulsive force?
  3. If you ar of a literary turn of mind, try you hand at using Japanese haiku (hi-koo), a form of poetry to summarize what you have learned so far in physics. The rules are quite simple: a haiku must have three lines, the first and third having five syllables and the second having seven syllables. No rhyming is necessary. Here is an example of a student haiku (by the way the plural is the same as the singular- haiku).

    Kepler took star dates,

    the shining stars in the sky,

    Physics came this way.

  4. Hold a mock trial for Copernicus. Two groups of students represent the prosecution and the defense. If possible have English, Social Studies, and language teachers serve as jurors for your trial.
  5. Read Berthold Brecht's play Galileo, and present part of it for the class. There is some controversy about whether the play truly reflects what historians believe were Galileo's feelings. For comparison, you could read The Crime of Galileo, by George de Santillana or Galileo and the Scientific Revolution, by Laura Fermi.
  6. Create a puzzle about astronomy. Present it to the class.

Credits: This experiment was first developed by Dr. Leo Lavatelli, University of Illinois, American Journal of Physics, vol.33, p605. The version used here is adapted from The Project Physics Course laboratory manual.

Your Assumptions | Effect of the Central Force | Scale the Plot | Computing Dv| Making the Plot | Prepare for Discussion | More Things to Do

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