Earth’s Tides Simulation

1. Earth’s Tides Simulation Craig Brubaker Union College 2/28/04

2. Agenda • Introduction • Purpose • Tidal Force • Tidal Bulges • Orbits • Simulation Overview • Inputs • Outputs • Example Simulation Configurations • Under the hood • Classes • Code Snapshots • Additional Work (features excluded)

3. Purpose of the Project To produce an application that helps students understand the tidal forces and the effects of orbital cycles on the tidal forces. To be used especially as a demonstration for introductory level courses. Simplifications of real system: • Assume Earth covered by a shallow ocean. • Parts of the system should be able to be turned on and off.

4. Tidal Force Recall Force of Gravity: • Proportional to the product of the masses over the square of the distance. Tidal Force: • Proportional to the product of the masses over the CUBE of the distance.

5. Laws of gravity and tides The gravitational attraction between two bodies is determined by Newton’s law of gravitation: A = gravitational attractive force. m1 and m2 = masses of the bodies attracting each other. r = distance between the centers of the two bodies. G = the gravitational constant. The tide generating force is derived from Newton’s law. This force is related to the difference between the gravitational attraction at Earth’s center and surface. F = the effective tidal force. µ = proportional to.

6. Major tidal influences on Earth Though much more massive than the moon, the sun is so much farther away that its tidal influence is less than half.

7. Tidal Bulge • Tidal Bulge • Tidal Force produces TWO high tide points. • Point on Earth closest to orbital body. • Point on Earth farthest from orbital body. • Tidal Force produces a low tide line equidistant to both high tide points. • The greater the high tide points, the lesser the low tide line. • Interaction of lunar and solar tides • At any point it is a simple sum.

8. Gravitational tidal bulge If we pretend that the moon does not orbit the Earth, gravitational attraction would raise a tidal bulge directed toward the moon. © Kurt Hollocher, 2002 The gravitational forces are directed exactly toward the moon’s center.

9. Earth-moon system: resultant forces on Earth’s surface The centrifugal and gravitational tidal vectors are shown resolved over the Earth’s surface. The vectors show that water is drawn into the tidal bulges, reaching maximum bulge elevation at points nearest and farthest from the moon. The lower diagram shows how, in a simple-minded model, two low and two high tides are expected daily.

10. Earth-sun-moon system: tide systematics Recall that the tidal bulges are identical on the sides of the Earth facing or opposite the moon. The same is true of the tides raised by the sun, except the solar tides are only ~46% as large as those of the lunar tides.

11. Orbits • Kepler’s Laws of Planetary Motion I. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.

12. Orbits • Kepler’s Laws of Planetary Motion II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.

13. Orbits • Kepler’s Laws of Planetary Motion III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes:

14. Ellipticity of Earth and moon orbits The orbits of the moon around Earth and of the Earth around the sun are ellipses. The Earth-moon orbital distance varies from 375,200 to 405,800 km, so the lunar tidal effect varies by 11.8% over a lunar month (inverse cube law, remember). The Earth-sun distance varies from 148,500,000 to 152,200,000 km, so the solar tidal effect varies by 3.7% over a year.

15. Simulation Overview • Inputs • EARTH ORBIT(in hours 0 to 8760): • 24 hours * 365 days • LUNAR ORBIT(in hours 0 to 708): • 24 hours * 29.5 days • Turn on/off (for Earth orbit and Lunar orbit) • Update orbits to entered orbit time values (again, for both) • Input sets • Advance by day or hour

16. Simulation Overview • Outputs • Tidal display • Displays resulting tide levels (tidal bulges) • Blue (low tide)  Green  Red (high tide) • %DISPLACEMENT • Orbiting body goes from a minimum distance to a maximum distance. • %DISPLACEMENT is the percent distance that the orbiting body is from its minimum distance. • Orbit times (for both)

17. Example(1a) Overlapping solar and lunar tides at maximum magnitudes

18. Example(1b) Overlapping solar and lunar tides at maximum magnitudes Turn OFF Earth orbit Update Earth

19. Example(1c) Overlapping solar and lunar tides at maximum magnitudes Turn OFF lunar orbit Update Moon

20. Example(2a) Solar and lunar tidal forces at right angles.

21. Example(2b) Solar and lunar tidal forces at right angles. Turn OFF Earth orbit

22. Example(2c) Solar and lunar tidal forces at right angles. Turn OFF lunar orbit

23. Example(3a) Input Set Two

24. Example(3b) Input Set Two ADVANCE DAY 3 times (or add 72 to orbit inputs and update each).

25. Example(3c) Input Set Two ADVANCE DAY 6 times (or add 144 to orbit inputs and update each).

26. Example(3d) Input Set Two ADVANCE DAY 9 times (or add 216 to orbit inputs and update each).

27. Under the hood • Java applet created in BlueJ:

28. Under the hood • Classes: • EarthTides (applet) • Buttons, labels, event handling • start method, paint method • Sun • Polar coordinates to Sun – geocentric: Earth at (0,0,0), rotate the Sun • Get magnitude of solar force • Advance function – geocentric: rotate the Sun • Moon • As Sun class, but for moon • Advance function: accounts for inclination of orbital plane • Pcoords • Polar coordinates: 2 directions and a distance • EarthPoint • Contains the directions that correspond to this point on the Earth and the magnitude of the tide. • Testing classes • TestSun, TestStart, TestPaint, RectTest.

29. Sun’s advance method (1) /** * advance method - advances the position of the Sun (geocentric) by hours * many hours. * * @param hours the amount of time to advance in hours */ public void advance(int hours) { double degMove = 360.0/(MAXTIMEEO + 1); // degrees to move each hour while ((hours > 0) && (orbitOn)) { // advance position for 1 hour if(timeEO == MAXTIMEEO) coords.dir1 = 0; // reset to correct small fraction that crops up else if( (coords.dir1 + degMove) < 360 ) coords.dir1 += degMove; else coords.dir1 = (coords.dir1 - 360) + degMove; coords.dist = MINDIST + (Math.sin(Math.toRadians((coords.dir1/2.0))) * (MAXDIST-MINDIST)); timeEO++; if(timeEO > MAXTIMEEO) timeEO = 0; hours--; } }

30. Sun’s advance method (2) • Approximation: for each time interval the change in angle that the Earth makes with the Sun (DegMove) is the same. • Accuracy? • The distance between the Earth and Sun is inversely proportional to the change in angle that the Earth makes with the Sun in a given time interval. • Recall Kepler’s 2nd law: equal areas swept in equal times. • MAXIMUM_DISTANCE = 152,104,980 km • MINIMUM_DISTANCE = 147,085,800 km. • % difference between distances = (MAXIMUM_DISTANCE - MINIMUM_DISTANCE /MAXIMUM_DISTANCE) * 100% = 3.29 %

31. Moon’s advance method /** * advance method - advances the position of the Sun (geocentric) by hours * many hours. * * @param hours the amount of time to advance in hours */ public void advance(int hours) { double degMove = 360.0/(MAXTIMEMO + 1); // degrees to move each hour while ((hours > 0) && (orbitOn)) { // advance position for 1 hour if(timeMO == MAXTIMEMO) coords.dir1 = 0; // reset to correct small fraction that crops up else if( (coords.dir1 + degMove) < 360 ) coords.dir1 += degMove; else coords.dir1 = (coords.dir1 - 360) + degMove; coords.dir2 = ORBITINCLINATION * Math.sin(Math.toRadians(coords.dir1)); if(coords.dir2 < 0) coords.dir2 += 360; coords.dist = MINDIST + (Math.sin(Math.toRadians((coords.dir1/2.0))) * (MAXDIST-MINDIST)); timeMO++; if(timeMO > MAXTIMEMO) timeMO = 0; hours--; } }

32. Part ofEarthTides start() method // determine which points to draw and assign corresponding directions pointSize = 5; // the size in pixels of each dimension for a point, // so pointSize = 5 represents 5x5 pixels double radius = 50; // circle's radius is radius # points long double i = 0; // holds a shifted coordinate value double j = 0; // holds an angle, which may need to be made positive for(int x = 0; x < 100; x++) for(int y = 0; y < 100; y++) { // for each possible point // if this point is part of the filled circle to draw // note: distance = sqrt(deltax^2 + deltay^2), // and circle center has coordinates (49.5,49.5) if(Math.sqrt(Math.pow(49.5-x,2) + Math.pow(49.5-y,2)) <= radius) { points[x][y].tideMag = 0; // include this one // calculate dir1: // leftmost x is 270 degrees, center is 0, rightmost is 90 i = x - 49.5; j = Math.toDegrees(Math.asin(i/49.5)); if (j >= 0) points[x][y].dir1 = j; else points[x][y].dir1 = 360 + j; // calculate dir2: // top y is 90, center is 0, bottom is 270 degrees i = y - 49.5; j = Math.toDegrees(Math.asin(i/49.5)); if (j > 0) points[x][y].dir2 = 360 - j; else points[x][y].dir2 = -1 * j; } else points[x][y].tideMag = -1; // discard this one }

33. EarthTides paint method (1) /** * paint - paints the tidal display area. * */ public void paint(Graphics g) { super.paint(g); // removes flicker effect for(int x = 0; x < 100; x++) for(int y = 0; y < 100; y++) { if(points[x][y].tideMag >= 0) { // if point is within filled circle // calculate tide magnitude for point[x][y] points[x][y].tideMag = 0; double deltaDir1 = 0; double deltaDir2 = 0; double angDist = 0;

34. EarthTides paint method (2) // solar tide // calculate change in each direction if (Math.abs(sun.coords.dir1 - points[x][y].dir1) <= 180) deltaDir1 = Math.abs(sun.coords.dir1 - points[x][y].dir1); else // look the other way deltaDir1 = 360 - Math.abs(sun.coords.dir1 - points[x][y].dir1); if (Math.abs(sun.coords.dir2 - points[x][y].dir2) <= 180) deltaDir2 = Math.abs(sun.coords.dir2 - points[x][y].dir2); else // look the other way deltaDir2 = 360 - Math.abs(sun.coords.dir2 - points[x][y].dir2); // currect for deltaDir's that are too large if(deltaDir1 >= 90) deltaDir1 = 180 - deltaDir1; if(deltaDir2 >= 90) deltaDir2 = 180 - deltaDir2; // angDist: angular distance to solar high tide angDist = Math.sqrt(Math.pow(deltaDir1,2) + Math.pow(deltaDir2,2)); // if angDist = 0 or 180, at a high tide point // if angDist = 90 or 270, at a low tide point points[x][y].tideMag += ( sun.getRelMag() * ((Math.cos(Math.toRadians(angDist*2)) + 1) / 2) );

35. EarthTides paint method (3) // lunar tide … similar to solar tide calculation // select color double mag = (int)points[x][y].tideMag; int red,green,blue = 0; if (mag < 500) { red = 0; blue = (int)(((500-mag)/500) * 255); green = (int)((mag/500) * 255); } else { red = (int)(((mag-500)/500) * 255); blue = 0; green = (int)(((1000/mag)-1) * 255); } theColor = new Color(red,green,blue); // draw the point g.setColor(theColor); g.fillRect(x*(pointSize), (y*(pointSize)) + OFFSET, pointSize, pointSize);

36. Additional Work • Moon’s precessions • Precession of the lunar orbital ellipse = 8.93 years. • Precession of the lunar orbital plane = 18.6 years. • Point Tracking • User selects a point on the Earth and the system dynamically displays a graph of the tidal level at that point. • Impact: until now, spin of the Earth not necessary. • The tidal bulge would look the same from a fixed view angle if the Earth spun or not. • Multiple View angles • Current simulation has only one.

37. The End.