Exploring the impact of Yukawa potential on Earth-Sun orbital parameters
In my recent exploration of celestial mechanics, I’ve explored the effects of the Yukawa potential on the Earth-Sun system. This potential modifies the classical Newtonian gravitational force by introducing an exponential decay term, providing a fresh perspective on the interaction between celestial bodies.
Introduction to Yukawa potential
The Yukawa potential is described by the equation:
U(r) = \frac{K_1}{r} e^{-\frac{\left\|\mathbf{r}_2 - \mathbf{r}_1\right\|}{K_2}}
where K_1 = GmM and K_2 is a scale length that introduces a decay in the gravitational interaction over distance. This potential allows us to simulate scenarios where gravity behaves differently at various scales, potentially mirroring conditions in exotic astronomical environments or under theoretical physics scenarios.
Modeling the Earth-Sun system
For this simulation, I used the known values of the Sun and Earth’s masses and their distance at aphelion and perihelion to establish initial conditions. Using these, I derived the expressions for the orbital energy and angular momentum, vital in maintaining the stability of the orbit under the new potential:
\begin{aligned} & E = -\frac{G M_{\odot} M_{\oplus}}{r_a + r_p} \\ & L = M_{\oplus} \sqrt{\frac{G M_{\odot} (2 r_p r_a)}{r_p + r_a}} \end{aligned}
The incorporation of the Yukawa potential requires recalculating these values at different points in the orbit to observe how the parameters change.
Computational analysis
I implemented a Python script to numerically solve for the new aphelion and perihelion under the Yukawa potential. The script utilizes the fsolve function from SciPy, which is an efficient tool for finding the roots of nonlinear equations.
import numpy as np
from scipy.optimize import fsolve
# Constants
M_sun = 1.989e30 # kg
M_earth = 5.972e24 # kg
G = 6.67430e-11 # m^3 kg^-1 s^-2
r_a = 1.47100396e+11 # m
r_p = 1.51854870e+11 # m
K1 = 2e15 # m
E = -G * M_sun * M_earth / (r_a + r_p)
L = M_earth * np.sqrt(G * M_sun * (r_a + r_p) / 2 * (
1 - ((r_a - r_p)**2 / (r_a + r_p)**2)))
def func_to_solve(r):
term1 = L**2 / (2 * M_earth * r**2)
term2 = -G * M_sun * M_earth / r * np.exp(-r / K1)
return term1 + term2 - E
# Initial guesses for the zeros
initial_guesses = [r_a, r_p]
# Finding the zeros
zeros = fsolve(func_to_solve, initial_guesses)
print("New aphelion and perihelion:", zeros)
Results and discussion
The recalculated orbital parameters indicate slight deviations from classical predictions, suggesting that even a small adjustment in gravitational theory could lead to observable changes in celestial mechanics. This finding underscores the sensitivity of orbital dynamics to the underlying gravitational model.
Conclusion
My analysis of the Yukawa potential’s impact on the Earth-Sun system underscores the need for precision in astronomical measurements and models. As our understanding of fundamental forces evolves, so too must our mathematical frameworks and computational approaches adapt to reflect these complexities accurately.
For more insights into this topic, you can find the details here.