КП: Движение спутника в двойной системе

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А.М. Кривцов > Теоретическая механика > Курсовые проекты ТМ 2015 > Движение спутника в двойной системе


Курсовой проект по Теоретической механике

Исполнитель: Мущак Никита

Группа: 09 (23604)

Семестр: весна 2015

Модель системы

Аннотация проекта[править]

Данный проект посвящен изучению движения спутника в двойной системе под действием гравитации. В ходе работы над проектом была написана программа, которая моделирует процесс движения спутника. Программа написана на языке JavaScript.

Формулировка задачи[править]

Исследовать движение спутника двойной системы под действием гравитационной силы. Двойная система состоит из 2 неподвижных планет и спутника вращающегося вокруг них как показано на рисунке сверху. Определить стационарные орбиты спутника, а также устойчивость движения спутника.

Общие сведения по теме[править]

Задачи подобного рода можно решать разными способами. Но решать данную задачу будем 2 способами :

с помощью уравнения Лагража 2-ого рода и как упрощенная задача 3-х тел

1 способ: уравнение Лагранжа 2-ого рода:

Lagrange.png




,где L - функция Лагранжа (лагранжиан),q- обобщенная координата, t — время, i— число степеней свободы механической системы

Функцию Лагранжа будем считать как разность кинетической и потенциальной энергий системы.

Дальнейшим дифференцированием получаем уравнение движения.

2 способ: записываем 2-ой закон Ньютона для данной задачи и получаем:

IC694010.png



, где G- гравитационная постоянная,m- массы планет.

Решение[править]

Ланранжиан будет иметь вид: LA.png, где m - масса спутника, q - обобщенная координата, Phi.png- потенциал гравитационного поля.

Подставляя полученное выражение в уравнение Лагранжа, можно получить уравнение движения: Equ.png

Как можно заметить из уравнения движения масса спутника никак не влияет на траекторию.

Отдельного рассмотрения заслуживает конфигурация потенциального гравитационного поля.

Оно будет иметь вид: Phi.jpg

При этом графики такого поля будут выглядеть:

Контурный график
Сравнение с овалами Кассини
3D график



































Стационарные орбиты спутника будут близки к овалам Кассини

-это семейство кривых, которые задаются уравнением Oval.png , где 2c-расстояние между фокусами, а- некоторая константа.

графики овалов Кассини: Cass.png



Частным случаем овалов Кассини является лемниската Бернулли, которая выглядит как знак бесконечности или восьмерка


Программа: скачать

Текст программы на языке JavaScript:

Файл "K3.html"

  1 <!DOCTYPE html>
  2 
  3 <html>
  4 <head>
  5   <meta charset="utf-8" />
  6   <meta http-equiv="X-UA-Compatible" content="IE=Edge" /> <!-- For IE on an intranet. -->
  7   <title>Moon in Binary System</title>
  8   <style>
  9     html, body {
 10       margin: 0;
 11       padding: 0;
 12     }
 13 
 14     html {
 15       overflow-y: scroll; /* There's an issue with the scrollbar "randomly" appearing - this just keeps it always visible in case the user is using a very wide and narrow monitor. */
 16     }
 17 
 18     body {
 19       width: 1024px; /* Currently, most screens can handle this. */
 20       margin: auto; /* Center the page content. */
 21       background-color: #777;
 22       font-family: "Segoe UI", Tahoma, Geneva, Verdana, sans-serif; /* Start screen font. */
 23     }
 24 
 25     header {
 26       color: #FFF;
 27       text-shadow: 5px 5px 10px #333;
 28     }
 29 
 30     section {
 31       position: relative; /* Float children relative to this element. */
 32     }
 33 
 34       section form {
 35         width: 210px; /* This is a bit less than the "section #WebGLCanvasElementContainer margin-left" value to provide a nice space between the form and the viewport. */
 36         float: left;
 37         text-align: center; /* Center the button elements. */
 38       }
 39 
 40         section form fieldset {
 41           text-align: left; /* Undo the button center aligning trick for the text in the form. */
 42           margin-bottom: 1.25em; /* Adjust this so that the height of the form is about the same height as the WebGL Three.js viewport element. */
 43         }
 44 
 45           section form fieldset input {
 46             width: 100%;
 47           }
 48 
 49         section form td {
 50           white-space: nowrap; /* Don't let words like "x-position" break at the hyphen (which occurs in Chrome). */
 51         }
 52 
 53       section #WebGLCanvasElementContainer {
 54         border: 1px solid #DDD; /* Match the native color of the fieldset border. */
 55         width: 800px; /* The assumed fixed width of the WebGL Three.js viewport element. */
 56         height: 600px; /* The assumed fixed height of the WebGL Three.js viewport element. */
 57         margin-left: 224px; /* This is "body width" minus "section #WebGLCanvasElementContainer width" or 1024px - 800px = 224px. */
 58         background-image: url('starField.jpg'); /* 0.15 opacity value. */
 59       }
 60 
 61       section article {
 62         padding: 0 1em;
 63         color: white;
 64       }
 65 
 66       section button {
 67         width: 4.5em;
 68       }
 69   </style>
 70   <script>
 71     /*// Preload all images/bitmaps.
 72     var preloadImages = [];
 73     var preloadImagePaths = ["jupiter.png", "saturn.png", "moon.png", "starField.jpg", "starField.jpg"];
 74     
 75     for (var i = 0; i < preloadImagePaths.length; i++) {
 76       preloadImages[i] = new Image();
 77       
 78       preloadImages[i].onerror = function() { 
 79         if (console) {
 80           console.error(this.src + " error.");
 81         } // if
 82       }; // onerror
 83       
 84       preloadImages[i].src = preloadImagePaths[i]; // Preload images to improve perceived app speed.
 85     } // for  
 86   */</script>
 87 </head>
 88 
 89 <body>
 90   <header>
 91     <h1>Moon in Binary System </h1>
 92   </header>
 93   <section>
 94     <form id="initialConditions">
 95       <fieldset>
 96         <legend>Moon</legend>
 97         <table id="mass1">
 98           <tr>
 99             <td>mass:</td>
100             <td><input id="m1_mass" type="number" value="1E18" required="required" /></td>
101           </tr>
102           <tr>
103             <td>x-position:</td>
104             <td><input id="m1_position_x" type="number" value="-141" required="required" /></td>
105           </tr>
106           <tr>
107             <td>y-position:</td>
108             <td><input id="m1_position_y" type="number" value="0" required="required" /></td>
109           </tr>
110           <tr>
111             <td>x-velocity:</td>
112             <td><input id="m1_velocity_x" type="number" value="0" required="required" /></td>
113           </tr>
114           <tr>
115             <td>y-velocity:</td>
116             <td><input id="m1_velocity_y" type="number" value="2" required="required" /></td>
117           </tr>
118           <tr style="display: none;">
119             <td>bitmap:</td>
120             <td><input type="text" value="moon.png" required="required" /></td>
121           </tr>
122         </table>
123       </fieldset>
124       <fieldset>
125         <legend>1st star</legend>
126         <table id="mass2">
127           <tr>
128             <td>mass:</td>
129             <td><input type="number" value="1E19" required="required" /></td>
130           </tr>
131           <tr>
132             <td>x-position:</td>
133             <td><input type="number" value="-100" required="required" /></td>
134           </tr>
135           <tr>
136             <td>y-position:</td>
137             <td><input type="number" value="0" required="required" /></td>
138           </tr>
139           <tr>
140             <td>x-velocity:</td>
141             <td><input type="number" value="0" required="required" /></td>
142           </tr>
143           <tr>
144             <td>y-velocity:</td>
145             <td><input type="number" value="0" required="required" /></td>
146           </tr>
147           <tr style="display: none;">
148             <td>bitmap:</td>
149             <td><input type="text" value="jupiter.png" required="required" /></td>
150           </tr>
151         </table>
152       </fieldset>
153       <fieldset>
154         <legend>2nd star</legend>
155         <table id="mass3">
156           <tr>
157             <td>mass:</td>
158             <td><input type="number" value="1E19" required="required" /></td>
159           </tr>
160           <tr>
161             <td>x-position:</td>
162             <td><input type="number" value="100" required="required" /></td>
163           </tr>
164           <tr>
165             <td>y-position:</td>
166             <td><input type="number" value="0" required="required" /></td>
167           </tr>
168           <tr>
169             <td>x-velocity:</td>
170             <td><input type="number" value="0"  required="required" /></td>
171           </tr>
172           <tr>
173             <td>y-velocity:</td>
174             <td><input type="number" value="0" required="required" /></td>
175           </tr>
176           <tr style="display: none;">
177             <td>bitmap:</td>
178             <td><input type="text" value="saturn.png" required="required" /></td>
179           </tr>
180         </table>
181       </fieldset>
182       <button id="submitButton">Submit</button>
183       <button id="reloadButton">Reload</button>
184       
185     </form>
186     <div id="WebGLCanvasElementContainer">
187       <!-- Three.js will add a canvas element to the DOM here. -->
188       <!-- The following <article> element (along with its content) will be removed via JavaScript just before the simulation starts: -->
189       <article>
190         <h2></h2>
191         <p>
192           
193         </p>
194         <h2>Running the simulation</h2>
195         <ul>
196           <li>To start the simulation with the current set of initial conditions, click the <strong>Submit</strong> button.</li>
197           <li>To orbit, left-click and drag the mouse.</li>
198           <li>To pan, right-click and drag the mouse.</li>
199           <li>To zoom, roll the mouse wheel.</li>
200           <li>To enter your own initial conditions, enter numeric values of your choice (in the form to the left) and click <strong>Submit</strong>. 
201           Note that large values such as 10<sup>18</sup> can be entered as 1E18.</li>
202           <li>To restart the simulation from scratch, click the <strong>Reload</strong> button (equivalent to refreshing the page).</li>
203           <li>For additional information and resources, click the <strong>Info</strong> button.</li>
204         </ul>
205       </article>
206     </div>
207   </section>
208   <script src="https://rawgithub.com/mrdoob/three.js/master/build/three.js"></script> <!-- The "CDN" for Three.js  -->
209   <script src="https://rawgithub.com/mrdoob/three.js/master/examples/js/controls/OrbitControls.js"></script> <!-- Allows for orbiting, panning, and zooming. -->
210   <script>
211     var  DENSITY= 1.38E14; // This value determined qualitatively by observing how large the spheres look onscreen (i.e., their radii).
212 
213     document.getElementById('submitButton').addEventListener('click', handleSubmitButton, false);
214     document.getElementById('reloadButton').addEventListener('click', handleReloadButton, false);
215     
216 
217     var simulation = Simulation(); // Call the Simulation constructor to create a new simulation object.
218 
219     function Simulation() { // A constructor.
220       var that = {}; // The object returned by this constructor.
221       var worker; // Will contain a reference to a fast number-chrunching worker thread that runs outside of this UR/animation thread.
222       var requestAnimationFrameID = null; // Used to cancel a prior requestAnimationFrame request.
223       var gl = {}; // Will contain WebGL related items.
224 
225       gl.viewportWidth = 800; // The width of the Three.js viewport.
226       gl.viewportHeight = 600; // The height of the Three.js viewport.
227 
228       gl.cameraSpecs = {
229         aspectRatio: gl.viewportWidth / gl.viewportHeight, // Camera frustum aspect ratio.
230         viewAngle: 50 // Camera frustum vertical field of view, in degrees.
231       };
232 
233       gl.clippingPlane = {
234         near: 0.1, // The distance of the near clipping plane (which always coincides with the monitor).
235         far: 1000 // The distance of the far clipping plane (note that you get a negative far clipping plane for free, which occurs at the negative of this value).
236       };
237 
238       gl.quads = 32; // Represents both the number of vertical segments and the number of horizontal rings for each mass's sphere wireframe.
239 
240       gl.renderer = window.WebGLRenderingContext ? new THREE.WebGLRenderer({ alpha: true }) : new THREE.CanvasRenderer({ alpha: true }); // If WebGL isn't supported, fallback to using the canvas-based renderer (which most browsers support). Note that passing in "{ antialias: true }" is unnecessary in that this is the default behavior. However, we pass in "{ alpha: true }" in order to let the background PNG image shine through.
241       gl.renderer.setClearColor(0x000000, 0); // Make the background completely transparent (the actual color, black in this case, does not matter) so that the PNG background image can shine through.
242       gl.renderer.setSize(gl.viewportWidth, gl.viewportHeight); // Set the size of the renderer.
243 
244       gl.scene = new THREE.Scene(); // Create a Three.js scene.
245 
246       gl.camera = new THREE.PerspectiveCamera(gl.cameraSpecs.viewAngle, gl.cameraSpecs.aspectRatio, gl.clippingPlane.near, gl.clippingPlane.far); // Set up the viewer's eye position.
247       gl.camera.position.set(0, 450, 0); // The camera starts at the origin, so move it to a good position.
248       gl.camera.lookAt(gl.scene.position); // Make the camera look at the origin of the xyz-coordinate system.
249 
250       gl.controls = new THREE.OrbitControls(gl.camera, gl.renderer.domElement); // Allows for orbiting, panning, and zooming via OrbitsControls.js by http://threejs.org. For an example, see http://threejs.org/examples/misc_controls_orbit.html.
251 
252       gl.pointLight = new THREE.PointLight(0xFFFFFF); // Set the color of the light source (white).
253       gl.pointLight.position.set(0, 250, 250); // Position the light source at (x, y, z).
254       gl.scene.add(gl.pointLight); // Add the light source to the scene.
255 
256       gl.spheres = []; // Will contain WebGL sphere mesh objects representing the point masses.
257 
258       var init = function (initialConditions) { // Public method, resets everything when called.
259         if (requestAnimationFrameID) {
260           cancelAnimationFrame(requestAnimationFrameID); // Cancel the previous requestAnimationFrame request.
261         }
262 
263         if (worker) {
264           worker.terminate(); // Terminate the previously running worker thread to ensure a responsive UI.
265         }
266         worker = new Worker('K3.js'); // Spawn a fast number-chrunching thread that runs outside of this UR/animation thread.
267 
268         document.getElementById('WebGLCanvasElementContainer').style.backgroundImage = "url('starField.jpg')"; // Switch back to the non-opaque PNG background image.
269         document.getElementsByTagName('article')[0].style.display = "none"; // Remove from page-flow the one (and only) article element (along with all of its content).
270         document.getElementById('WebGLCanvasElementContainer').appendChild(gl.renderer.domElement); // Append renderer element to DOM.
271 
272         while (gl.spheres.length) { // Remove any prior spheres from the scene and empty the gl.spheres array:
273           gl.scene.remove(gl.spheres.pop());
274         } // while
275 
276         for (var i = 0; i < initialConditions.length; i++) { // Set the sphere objects in gl.spheres to initial conditions.
277           initializeMesh(initialConditions[i]); // This call sets the gl.spheres array.
278         } // for
279 
280         worker.postMessage({
281           cmd: 'init', // Pass the initialization command to the web worker.
282           initialConditions: initialConditions // Send a copy of the initial conditions to the web worker, so it can initialize its persistent global variables.
283         }); // worker.postMessage
284 
285         worker.onmessage = function (evt) { // Process the results of the "crunch" command sent to the web worker (via this UI thread).
286           for (var i = 0; i < evt.data.length; i++) {
287             gl.spheres[i].position.x = evt.data[i].p.x;
288             gl.spheres[i].position.z = evt.data[i].p.y;
289             gl.spheres[i].position.y = 0; // 3BodyWorker.js is 2D (i.e., the physics are constrained to a plane).
290             gl.spheres[i].rotation.y += initialConditions[i].rotation; // Place worker.onmessage in the init method in order to access its initialConditions array.
291           }
292           gl.renderer.render(gl.scene, gl.camera); // Update the positions of the masses (sphere meshes) onscreen based on the data returned by 3BodyWorker.js.
293         }; // worker.onmessage
294 
295         function initializeMesh(initialCondition) {
296           var texture = THREE.ImageUtils.loadTexture(initialCondition.bitmap); // Create texture object based on the given bitmap path.
297           var material = new THREE.MeshPhongMaterial({ map: texture }); // Create a material (for the spherical mesh) that reflects light, potentially causing sphere surface shadows.
298           var geometry = new THREE.SphereGeometry(initialCondition.radius, gl.quads, gl.quads); // Radius size, number of vertical segments, number of horizontal rings.
299           var mesh = new THREE.Mesh(geometry, material); // A mesh represents the object (typically composed of many tiny triangles) to be displayed - in this case a hollow sphere with a bitmap on its surface.
300 
301           mesh.position.x = initialCondition.position.x;
302           mesh.position.z = initialCondition.position.y; // Convert from 2D to "3D".
303           mesh.position.y = 0; // The physics are constrained to the xz-plane (i.e., the xy-plane in 3BodyWorker.js).
304 
305           gl.scene.add(mesh); // Add the sphere to the Three.js scene.
306           gl.spheres.push(mesh); // Make the Three.js mesh sphere objects accessible outside of this helper function.
307         } // initializeMesh
308       } // init
309       that.init = init; // This is what makes the method public.
310 
311       var run = function () { // Public method.
312         worker.postMessage({
313           cmd: 'crunch' // This processing occurs between animation frames and, therefore, is assumed to take a relatively small amount of time (as compared to current frame rates).
314         }); // worker.postMessage
315         gl.controls.update(); // Allows for orbiting, panning, and zooming.
316         requestAnimationFrameID = requestAnimationFrame(run); // Allow for the cancellation of this requestAnimationFrame request.
317       }; // run()
318       that.run = run;
319 
320       return that; // The object returned by the constructor.
321     } // Simulation
322 
323     function handleSubmitButton(evt) {
324       var m1 = InitialCondition(document.getElementById('mass1').querySelectorAll('input')); // A constructor returning an initial condition object.
325       var m2 = InitialCondition(document.getElementById('mass2').querySelectorAll('input'));
326       var m3 = InitialCondition(document.getElementById('mass3').querySelectorAll('input'));
327 
328       evt.preventDefault(); // Don't refresh the page when the user clicks this form button.
329 
330       if (!window.WebGLRenderingContext) { displayCanvasRendererWarning(); } // If necessary, warn the user that they're using a canvas-based Three.js renderer and that they should upgrade their browser so that a faster WebGL-based renderer can be used instead.
331 ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
332       simulation.init([m1, m2, m3]);
333 ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////     
334 	 simulation.run(); // The images have been preloaded so this works immediately.
335 
336       function InitialCondition(inputElements) {
337         var mass = parseFloat(inputElements[0].value);
338 
339         return {
340           mass: mass,
341           radius: calculateRadius(mass),
342           rotation: calculateRotation(mass),
343           position: { x: parseFloat(inputElements[1].value), y: parseFloat(inputElements[2].value) },
344           velocity: { x: parseFloat(inputElements[3].value), y: parseFloat(inputElements[4].value) },
345           bitmap: inputElements[5].value // This is a string value (hence the non-use of parseFloat).
346         };
347 
348         function calculateRadius(mass) {
349           /*
350             Mass equals density times volume or m = D * V = D * (4/3 * PI * r^3), and solving for r = [(3 * m)/(4 * PI * D)]^(1/3)
351           */
352           var radicand = (3 * mass) / (4 * Math.PI * DENSITY); // Only change the value of DENSITY to affect the value returned by this function.
353 
354           return Math.pow(radicand, 1 / 3);
355         } // calculateRadius
356 
357         function calculateRotation(mass) {
358           /*
359             Using a power model, let the x-axis represent the radius and the y-axis the rotational rate of the sphere. 
360             The power model is y = a * x^b, where a and b are constants (which were empirically derived).
361           */
362           var radius = calculateRadius(mass);
363 
364           return 1.7 * Math.pow(radius, -1.9); // Rotational rate as a function of the sphere's radius.
365         } // calculateRotation
366       } // InitialCondition
367     } // handleSubmitButton
368 
369     function handleReloadButton(evt) {
370       /*  
371         Clicking a form button automatically refreshes the page, which is exactly the behavior we want (i.e., location.reload() is not necessary here).
372       */
373     } // handleReloadButton
374 
375     function handleInfoButton(evt) {
376       /*
377         Note that when the info page covers up the animation, the animation stops because this is how requestAnimationFrame works. In this sense, we get a free pause feature.
378       */
379       evt.preventDefault(); // Don't refresh the page when the user clicks this form button.
380       window.open("info.html"); // Open the info.html page in another tab.
381     } // handleInfoButton
382 
383     function displayCanvasRendererWarning() { // This assumes that the user's browser at least supports canvas.
384       var articleElement = document.getElementsByTagName('article')[0];
385 
386       articleElement.innerHTML = "<h2>WebGL not supported, using canvas-based renderer, please upgrade your browser.</h2>";
387       articleElement.style.display = "block";
388     }
389   </script>
390 </body>
391 </html>


Текст программы на языке JavaScript (продолжение):

Файл "K3.js"

  1 /* 
  2 The acceleration equations for the 2D three-body problem (see equations 42 through 50):
  3 
  4   d^2[x1]/dt^2 = G*m2*(x2 - x1)/alpha + G*m3*(x3 - x1)/beta
  5 
  6   d^2[y1]/dt^2 = G*m2*(y2 - y1)/alpha + G*m3*(y3 - y1)/beta
  7 
  8   d^2[x2]/dt^2 = G*m1*(x1 - x2)/alpha + G*m3*(x3 - x2)/gamma
  9 
 10   d^2[y2]/dt^2 = G*m1*(y1 - y2)/alpha + G*m3*(y3 - y2)/gamma
 11 
 12   d^2[x3]/dt^2 = G*m1*(x1 - x3)/beta  + G*m2*(x2 - x3)/gamma
 13 
 14   d^2[y3]/dt^2 = G*m1*(y1 - y3)/beta  + G*m2*(y2 - y3)/gamma
 15 
 16 where G = gravitational constant = 6.6725985 X 10^(-11) N-m^2/kg^2
 17       alpha = [ (x2 - x1)^2 + (y2 - y1)^2 ]^(3/2)  AND  alpha <> 0
 18       beta  = [ (x1 - x3)^2 + (y1 - y3)^2 ]^(3/2)  AND  beta <> 0
 19       gamma = [ (x3 - x2)^2 + (y3 - y2)^2 ]^(3/2)  AND  gamma <> 0
 20 */
 21 
 22 var N = 3; // The number of bodies (point masses) this code is designed to handle.
 23 var G = 6.67384E-11; // Big-G, in N(m/kg)^2.
 24 var h = 0.000001; // Interval between time steps, in seconds. The smaller the value the more accurate the simulation. This value was empirically derived by visually observing the simulation over time.
 25 var iterationsPerFrame = 400; // The number of calculations made per animation frame, this is an empirically derived number based on the value of h.
 26     
 27 var m1;
 28 var m1_half; // Initially, will contain a copy of m1.
 29 var m2;
 30 var m2_half; 
 31 var m3;
 32 var m3_half;
 33 
 34 self.onmessage = function (evt) { // evt.data contains the data passed from the calling main page thread.
 35   switch (evt.data.cmd) {
 36     case 'init':
 37       init(evt.data.initialConditions); // Transfer the initial conditions data to the persistant variables in this thread.
 38       break;
 39     case 'crunch':
 40       crunch();
 41       break;
 42     default:
 43       console.error("ERROR FROM worker.js: SWITCH STATEMENT ERROR IN self.onmessage");
 44   } // switch
 45 };
 46 
 47 // The denominators alpha, beta, and gamma for the acceleration equations 42 through 47:
 48 function alpha(m1, m2) { // Equation 48.
 49   var delta_x = m2.p.x - m1.p.x;
 50   var delta_y = m2.p.y - m1.p.y;
 51 
 52   var delta_x_squared = delta_x * delta_x;
 53   var delta_y_squared = delta_y * delta_y;
 54 
 55   var base = delta_x_squared + delta_y_squared;
 56 
 57   return Math.sqrt(base * base * base); // Raise the base to the 3/2 power so as to calculate (x_2 - x_1 )^2 + (y_2 - y_1 )^2]^(3/2), equation 48.
 58 }
 59 
 60 function beta(m1, m3) { // Equation 49.
 61   var delta_x = m3.p.x - m1.p.x;
 62   var delta_y = m3.p.y - m1.p.y;
 63 
 64   var delta_x_squared = delta_x * delta_x;
 65   var delta_y_squared = delta_y * delta_y;
 66 
 67   var base = delta_x_squared + delta_y_squared;
 68 
 69   return Math.sqrt(base * base * base); // Raise the base to the 3/2 power so as to calculate (x3 - x1)^2 + (y3 - y1)^2 ]^(3/2), equation 49.
 70 }
 71 
 72 function gamma(m2, m3) { // Equation 50.
 73   var delta_x = m3.p.x - m2.p.x; 
 74   var delta_y = m3.p.y - m2.p.y;
 75 
 76   var delta_x_squared = delta_x * delta_x;
 77   var delta_y_squared = delta_y * delta_y;
 78 
 79   var base = delta_x_squared + delta_y_squared;
 80 
 81   return Math.sqrt(base * base * base); // Raise the base to the 3/2 power so as to calculate (x3 - x2)^2 + (y3 - y2)^2]^(3/2), equation 50.
 82 }
 83 
 84 /*
 85   Note that the alpha, beta, and gamma functions could be replaced with a single alpha_beta_gamma(massA, massB) function but for clarity, this was not done.
 86 */
 87 
 88 this.init = function (initialConditions) {
 89 
 90   // Define local mass object constructor function:
 91   function Mass(initialCondition) {
 92     this.m = initialCondition.mass; // The mass of the point mass.
 93     this.p = { x: initialCondition.position.x, y: initialCondition.position.y }; // The position of the mass.
 94     this.v = { x: initialCondition.velocity.x, y: initialCondition.velocity.y }; // The x- and y-components of velocity for the mass.
 95     this.a = {}; // Will contain the x- and y-components of acceleration for the mass.
 96   }
 97 
 98   if (initialConditions.length != N) {
 99     console.error("ERROR FROM worker.js: THE initialConditions ARRAY DOES NOT CONTAIN EXACTLY " + N + " OBJECTS - init() TERMINATED");
100     return;
101   }
102 
103   // Set the local mass object global variables:
104   m1 = new Mass(initialConditions[0]);
105   m1_half = new Mass(initialConditions[0]); // Create a copy of m1.
106   m2 = new Mass(initialConditions[1]);
107   m2_half = new Mass(initialConditions[1]); 
108   m3 = new Mass(initialConditions[2]);
109   m3_half = new Mass(initialConditions[2]); 
110 
111   // Calculate initial acceleration values (using initial conditions) in preparation for using equation 25:
112   m1.a.x = G * m2.m * (m2.p.x - m1.p.x) / alpha(m1, m2) + G * m3.m * (m3.p.x - m1.p.x) / beta(m1, m3); // Equation 42.
113   m1.a.y = G * m2.m * (m2.p.y - m1.p.y) / alpha(m1, m2) + G * m3.m * (m3.p.y - m1.p.y) / beta(m1, m3); // Equation 43.
114   m2.a.x = G * m1.m * (m1.p.x - m2.p.x) / alpha(m1, m2) + G * m3.m * (m3.p.x - m2.p.x) / gamma(m2, m3); // Equation 44.
115   m2.a.y = G * m1.m * (m1.p.y - m2.p.y) / alpha(m1, m2) + G * m3.m * (m3.p.y - m2.p.y) / gamma(m2, m3); // Equation 45.
116   m3.a.x = G * m1.m * (m1.p.x - m3.p.x) / beta(m1, m3)  + G * m2.m * (m2.p.x - m3.p.x) / gamma(m2, m3); // Equation 46.
117   m3.a.y = G * m1.m * (m1.p.y - m3.p.y) / beta(m1, m3)  + G * m2.m * (m2.p.y - m3.p.y) / gamma(m2, m3); // Equation 47.
118 
119   function equation25(x, v, a) {
120     return x + 0.5 * h * v + 0.25 * (h * h) * a;  // Equation 25.
121   }
122 
123   // For the first iteration (and only the first iteration), use equation 25 (instead of equation 22) to calculate the initial half-integer position values:
124   m1_half.p.x = equation25(m1.p.x, m1.v.x, m1.a.x);
125   m1_half.p.y = equation25(m1.p.y, m1.v.y, m1.a.y);
126   m2_half.p.x = equation25(m2.p.x, m2.v.x, m2.a.x);
127   m2_half.p.y = equation25(m2.p.y, m2.v.y, m2.a.y);
128   m3_half.p.x = equation25(m3.p.x, m3.v.x, m3.a.x);
129   m3_half.p.y = equation25(m3.p.y, m3.v.y, m3.a.y);
130 } // this.init
131 
132 
133 this.crunch = function () {
134   for (var i = 0; i < iterationsPerFrame; i++) {
135     // Calculate half-integer acceleration values (using equations 18 through 21) in preparation for using equation 23:
136     m1_half.a.x = G * m2_half.m * (m2_half.p.x - m1_half.p.x) / alpha(m1_half, m2_half) + G * m3_half.m * (m3_half.p.x - m1_half.p.x) / beta(m1_half, m3_half); // Equation 42.
137     m1_half.a.y = G * m2_half.m * (m2_half.p.y - m1_half.p.y) / alpha(m1_half, m2_half) + G * m3_half.m * (m3_half.p.y - m1_half.p.y) / beta(m1_half, m3_half); // Equation 43.
138     m2_half.a.x = G * m1_half.m * (m1_half.p.x - m2_half.p.x) / alpha(m1_half, m2_half) + G * m3_half.m * (m3_half.p.x - m2_half.p.x) / gamma(m2_half, m3_half); // Equation 44.
139     m2_half.a.y = G * m1_half.m * (m1_half.p.y - m2_half.p.y) / alpha(m1_half, m2_half) + G * m3_half.m * (m3_half.p.y - m2_half.p.y) / gamma(m2_half, m3_half); // Equation 45.
140     m3_half.a.x = G * m1_half.m * (m1_half.p.x - m3_half.p.x) / beta(m1_half, m3_half)  + G * m2_half.m * (m2_half.p.x - m3_half.p.x) / gamma(m2_half, m3_half); // Equation 46.
141     m3_half.a.y = G * m1_half.m * (m1_half.p.y - m3_half.p.y) / beta(m1_half, m3_half)  + G * m2_half.m * (m2_half.p.y - m3_half.p.y) / gamma(m2_half, m3_half); // Equation 47.
142     
143     // Calculate velocity values using equation 23:
144     m1.v.x = equation23(m1.v.x, m1_half.a.x);
145     m1.v.y = equation23(m1.v.y, m1_half.a.y);
146     m2.v.x = equation23(m2.v.x, m2_half.a.x);
147     m2.v.y = equation23(m2.v.y, m2_half.a.y);
148     m3.v.x = equation23(m3.v.x, m3_half.a.x);
149     m3.v.y = equation23(m3.v.y, m3_half.a.y);
150     
151     // Calculate position values using equation 24:
152     m1.p.x = equation24(m1_half.p.x, m1.v.x);
153     m1.p.y = equation24(m1_half.p.y, m1.v.y);
154     m2.p.x = equation24(m2_half.p.x, m2.v.x);
155     m2.p.y = equation24(m2_half.p.y, m2.v.y);
156     m3.p.x = equation24(m3_half.p.x, m3.v.x);
157     m3.p.y = equation24(m3_half.p.y, m3.v.y);
158 
159     // Calculate half-integer position values using equation 22:
160     m1_half.p.x = equation22(m1.p.x, m1.v.x);
161     m1_half.p.y = equation22(m1.p.y, m1.v.y);
162     m2_half.p.x = equation22(m2.p.x, m2.v.x);
163     m2_half.p.y = equation22(m2.p.y, m2.v.y);
164     m3_half.p.x = equation22(m3.p.x, m3.v.x);
165     m3_half.p.y = equation22(m3.p.y, m3.v.y);
166   } // for
167 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
168   self.postMessage([m1/*, m2, m3*/]); // Send the crunched data back to the UI thread to be rendered onscreen.
169 //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
170   function equation23(v, a) {
171     return v + h * a; // Equation 23.
172   }
173 
174   function equation24(x, v) {
175     return x + 0.5 * h * v; // Equation 24.
176   }
177 
178   function equation22(x, v) {
179     return x + 0.5 * h * v; // Equation 22, this function is of course the same as the equation24(x, v) function.
180   }
181 } // this.crunch

В случае трех материальных тел на каждое из них действуют две силы со стороны двух других тел. Например, на тело m₁ действуют следующие силы (F₂ и F₃): IC694.png




Сначала заметим, что результирующая сила F₁, действующая на тело m₁, будет суммой сил F₂ и F₃. Это значит, что F₁ = m₁a₁ = F₂ + F₃. Теперь по тригонометрическим законам, мы можем разложить модуль результирующей силы F₁, действующей на тело m₁, на компоненты x и y: IC694007.png

В красном и зеленом треугольниках на рис. мы видим:

IC694008.png




Согласно закону всемирного тяготения Ньютона, F₂ и F₃ можно выразить как: IC694009.png



Подставляя формулы, получим:

IC694010.png


Упрощая формулы, имеем: IC694012.png


Здесь α и β равны: IC694013.png

Эту систему уравнений (34,35,38-41) можно решить численно методом интегрирования "чехарда" (формулы 22–24) по заданным начальным условиям (значения массы, положения и скорости для каждого тела) с приемлемой точностью и стабильностью. Чтобы быстро добиться высокой точности, можно использовать рабочий веб-процесс для выполнения численного интегрирования в потоке, отдельном от потока пользовательского интерфейса главной страницы.

Рассмотрим N небесных тел. Пусть i обозначает одно из тел (i = 1, …, N), а h — малый интервал времени. В позиционном алгоритме Верле следующие значения положения и скорости тела i вычисляются следующим образом: IC693998.png

номера возле формул соответствуют номерам формул в программе(см.текст программы K3.js)

Обсуждение результатов и выводы[править]

Таким образом, в ходе работы над проектом, смоделирован процесс движения спутника в двойной системе. Обработка результатов показала, что спутник пролетает вокруг двух тел по лемнискате один раз, а потом отклоняется. Это отклонение получается за счет накопления ошибки вычислений в программе.


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