Mathematics Helps James Webb Space Telescope

Two Sides of James Webbin
Two Sides of James Webbin

In the final days of 2021, mathematicians wrote the first equations describing the location of the James Webb Space Telescope. Webb will stay in the cosmic parking lot for about 20 years, studying the galaxies of the universe. And we don't have to worry about it getting lost.

Its new home is the Lagrangian point, a gravitationally stable position with respect to the Earth and the sun.

At Lagrange point 2 (L2), one of these five points in the Sun-Earth system, Webb perceives the gravitational pull on him from both our planet and the sun.

The centripetal force that causes objects to move in a circle around a gravitational object also pushes the telescope into orbit with this system, causing it to spin around and pull towards L2.

Lagrangian points are popular with space researchers because they stay in fixed places when viewed from Earth, making them useful for communicating with spacecraft. In the 18th century, mathematicians like Webb identified five Lagrangian points that govern the motion of moons. This was an exercise in understanding the motions of a two-body system such as the Earth and the Moon.

According to astrophysicist Neil Cornish, this meant an infinite number of solutions to the three-mass problem.
You should use Newton's second law of motion to calculate the total force exerted on the smaller mass object; this states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. When you push an empty shopping cart against a full one, you will notice that the full cart moves slower and requires more force to push.

However, you cannot ignore the movements of all three bodies. The Earth rotates on its axis, causing the Coriolis effect, which causes objects to move in curved lines (This is why hurricanes and bullets follow a curved path). Centripetal force also causes an object revolving around a central mass to be pulled towards the center of that mass.

Cornish considered symmetry along a line of two points with the sun and Earth on either side. This logic excludes any Lagrangian points outside the ecliptic plane (the imaginary plane containing Earth's orbit around the sun) and the Earth on either side.

This logic excludes any Lagrangian point (the imaginary plane containing Earth's orbit around the sun) outside the plane of the ecliptic. Three Lagrangian points (L1, L2 and L3) are unstable and lie along this line, while two are stable and symmetrical (L4 and L5) and lie as points of an equilateral triangle above and below this line.
“I was able to eliminate an entire class of solutions with just a little thought,” Cornish says, “rather than just diving in and using brute force.”
Mathematics is added to the mix to describe the stability of each point, which is critical for sending space missions to Lagrangian points.”

Calculus shakes the model to determine whether forces will hold an object in place or move it away over time.
If the Lagrangian point is not completely fixed, as with the Webbs, the spacecraft must make regular course corrections with a small burn of fuel to return to the center of the point. Webb's fuel will run out in about 20 years and it will move away from L2. Cornish believes it will leave our solar system and become an interstellar rover.
Want to try your hand at finding Lagrangian points? A student with an undergraduate degree in advanced mechanics and vector algebra has all the tools necessary to find these solutions.

Let's simplify the concept of Lagrangian points. Consider a bowling ball (the sun) and a baseball (Earth) in a horizontal plane, each with its own gravitational pull. Bowling ball has a stronger overall pull than baseball because it is much heavier. After that throw a marble (satellite) between the two. If properly balanced between two depressions, it's like being in a "saddle point" between the gravity well of two large objects in the plane. However, if you push the marble too far in either direction, it will be attracted by the larger object.

Source: Popular Mechanics

📩 01/05/2022 15:33

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