Orbital Trajectory Calculator
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Orbital Trajectory Calculator
Orbital Parameters
Enter parameters above to see results.
Understanding Orbital Trajectories
An orbital trajectory describes the path an object takes around another object in space due to gravitational force. This calculator estimates key orbital parameters based on initial conditions, providing insights into celestial mechanics. It utilizes a simplified numerical integration method (like the Euler or Verlet method) to approximate the path over time, considering the gravitational interaction between two bodies.
The calculation is based on Newton's Law of Universal Gravitation and basic kinematics. The force of gravity between two masses, $m_1$ and $m_2$, separated by a distance $r$, is given by:
$F = G \frac{m_1 m_2}{r^2}$
Where $G$ is the gravitational constant ($6.67430 \times 10^{-11} \text{ N m}^2/\text{kg}^2$). This force causes acceleration on both bodies. For simplicity in many calculators, we often focus on the acceleration of a smaller body around a much larger one, where the larger body's motion is negligible:
$a = \frac{F}{m_2} = G \frac{m_1}{r^2}$
The calculator simulates the trajectory by:
- Calculating the gravitational force and acceleration at the current position.
- Updating the velocity based on this acceleration over a small time step ($\Delta t$).
- Updating the position based on the new velocity over the same time step.
- Repeating this process for the specified total simulation time.
The initial velocity vector and launch angle determine the shape of the orbit (e.g., circular, elliptical, parabolic, or hyperbolic). A 90-degree angle typically implies an orbit directly away from or towards the primary body, or a tangential orbit if combined with appropriate velocity.
Use Cases:
- Astronomy Education: Demonstrating basic orbital mechanics principles.
- Mission Planning (Conceptual): Providing a rough estimate of spacecraft trajectories.
- Science Fiction World-Building: Understanding the gravitational dynamics of fictional systems.
- Hobbyist Astrophysics: Exploring the relationship between mass, distance, and velocity in orbital systems.
Note: This is a simplified model. Real-world orbital calculations often involve more complex factors like the gravitational influence of other celestial bodies, atmospheric drag (if applicable), relativistic effects, and non-uniform mass distributions. The accuracy depends heavily on the chosen time step and total simulation duration. A smaller time step generally increases accuracy but requires more computation.
var G = 6.67430e-11; // Gravitational constant
function calculateOrbitalTrajectory() {
var mass1 = parseFloat(document.getElementById("mass1").value);
var mass2 = parseFloat(document.getElementById("mass2").value);
var r = parseFloat(document.getElementById("distance").value); // Initial distance
var v_initial = parseFloat(document.getElementById("velocity").value); // Initial speed
var angle_degrees = parseFloat(document.getElementById("angle").value);
var dt = parseFloat(document.getElementById("timeStep").value); // Time step
var totalTime = parseFloat(document.getElementById("simulationTime").value); // Total simulation time
var resultDiv = document.getElementById("result");
// Input validation
if (isNaN(mass1) || isNaN(mass2) || isNaN(r) || isNaN(v_initial) || isNaN(angle_degrees) || isNaN(dt) || isNaN(totalTime)) {
resultDiv.textContent = "Error: Please enter valid numerical values for all fields.";
return;
}
if (mass1 <= 0 || mass2 <= 0) {
resultDiv.textContent = "Error: Masses must be positive.";
return;
}
if (r <= 0) {
resultDiv.textContent = "Error: Initial distance must be positive.";
return;
}
if (dt <= 0 || totalTime <= 0) {
resultDiv.textContent = "Error: Time step and total simulation time must be positive.";
return;
}
// Convert angle to radians
var angle_rad = angle_degrees * Math.PI / 180.0;
// Initial position and velocity vectors (assuming primary body is at origin)
// We'll simplify by considering the 2D plane defined by the initial velocity vector and the line connecting the bodies.
// var initial position be (r, 0) and initial velocity components depend on the angle.
var pos_x = r;
var pos_y = 0;
var vel_x = v_initial * Math.cos(angle_rad);
var vel_y = v_initial * Math.sin(angle_rad);
var currentTime = 0;
var trajectoryPoints = [{x: pos_x, y: pos_y}];
// Numerical integration loop (simplified Euler method for demonstration)
while (currentTime 100000) {
resultDiv.textContent = "Simulation stopped: Exceeded maximum points. Orbit might be unstable or too long.";
return;
}
}
// Basic analysis of the simulated trajectory
var final_pos_x = trajectoryPoints[trajectoryPoints.length – 1].x;
var final_pos_y = trajectoryPoints[trajectoryPoints.length – 1].y;
var final_r = Math.sqrt(final_pos_x*final_pos_x + final_pos_y*final_pos_y);
var final_vel_x = vel_x; // Last calculated velocity
var final_vel_y = vel_y;
var final_v = Math.sqrt(final_vel_x*final_vel_x + final_vel_y*final_vel_y);
var analysis = "Simulation complete. ";
if (final_r 10) { // Simplified collision check
analysis += "Potential collision or close approach detected. ";
}
// Determine orbit type based on energy (more accurate methods exist, this is a simplification)
// Kinetic Energy per unit mass: 0.5 * v^2
// Potential Energy per unit mass: – G * M_primary / r
var kineticEnergyPerMass = 0.5 * final_v * final_v;
var potentialEnergyPerMass = -G * mass1 / final_r;
var totalEnergyPerMass = kineticEnergyPerMass + potentialEnergyPerMass;
var orbitType = "Undetermined";
var escapeVelocity = Math.sqrt(2 * G * mass1 / final_r);
if (totalEnergyPerMass < 0) {
// Bound orbit (Elliptical or Circular)
if (Math.abs(kineticEnergyPerMass – (-potentialEnergyPerMass)) < 0.1 * Math.abs(totalEnergyPerMass) ) {
orbitType = "Approximately Circular";
} else {
orbitType = "Elliptical";
}
} else if (totalEnergyPerMass === 0) { // Highly unlikely with floats
orbitType = "Parabolic";
} else {
// Unbound orbit (Hyperbolic)
orbitType = "Hyperbolic";
}
resultDiv.innerHTML = analysis +
"
Final Distance: " + final_r.toExponential(3) + " m" +
"
Final Speed: " + final_v.toExponential(3) + " m/s" +
"
Estimated Orbit Type: " + orbitType +
"(Based on simplified simulation)";
// Optional: Further analysis could include calculating semi-major axis, eccentricity, period etc.
// This requires more complex orbital mechanics formulas depending on the orbit type.
}