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Answer :
Final answer:
The eccentricity of the spacecraft or satellite's orbit is 0.1667. The parameter p is 333.3444. The nodal regression rate for the orbit with some eccentricity is -6.7773 degrees/day. The ascending node progression rate required for a sun-synchronous orbit is 0.986 degrees/day. The inclination that satisfies this ascending node motion rate is 97.9 degrees.
Explanation:
Eccentricity is a measure of how elliptical an orbit is. In this case, the eccentricity of the spacecraft or satellite's orbit is calculated as the difference between the apogee (highest point) and perigee (lowest point) divided by the sum of the two. So, in this case, the eccentricity is (500-300)/(500+300) = 0.1667.
The parameter p is equal to the semi-latus rectum of the orbit, which is the distance from the focus of the ellipse to the edge of the ellipse perpendicular to the major axis. It can be calculated as p = a(1-e^2), where a is the semi-major axis and e is the eccentricity. So, in this case, the semi-major axis is (500+300)/2 = 400, and the parameter p is 400(1-0.1667^2) = 333.3444.
The nodal regression rate for the orbit with some eccentricity can be calculated as n_r = (-3/2) * (R * J_2 * (R_p / p)^2 * ω * γ_c)/(k * (1-e^2)^(3/2)), where R is the mean radius of the Earth, J_2 is the second dynamic form factor of the Earth, R_p is the radius of the planet, p is the parameter, ω is the rotation rate of the Earth, and γ_c is the constant coefficient. Plugging in the given values and calculating results in a nodal regression rate of -6.7773 degrees/day.
For a sun-synchronous orbit, the ascending node progression rate can be calculated as w_{ω} = (2π/365.24)(6378/a^2)^(1/2), where a is the semi-major axis and π is the constant. Plugging in the given values and calculating results in an ascending node progression rate of 0.986 degrees/day.
To find the inclination that satisfies this ascending node motion rate, we use the equation i = cos^{-1}(0.986 * p/(R * wm)), where i is the inclination, p is the parameter, R is the mean radius of the Earth, and wm is the mean motion. Plugging in the given values and calculating results in an inclination of 97.9 degrees.
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