(1.1) Perturbed two-body problem Consider Earth-orbiting spacecraft. Define Earth-centered Inertial (ECI) frame N: {0, 1, 2, 3} with its origin at the Earth center and having 3 aligned with the Earth rotation north pole. Let r = ïñ₁+y^2+zñ³ and v = v₁₁+v₁n₂+v₂n3 represent the position and velocity vectors of the spacecraft orbit, respectively. We consider the perturbation due to the Earth irregular gravity field (J2 only) acting on the orbit: a12 215 _342 [(1 [(1-5) Th xn₁ + 1+ (1-5) y yn2+3 +(3-5) zn] (1) where r = ||||2, and we take the Earth's equatorial radius as the normalizing radius, ro. Specific values for these parameters assumed for this problem is summarized in Table 1. Table 1: Assumed dynamical parameter values (Problem (1.1)) parameter Symbol Value Unit Earth gravitational parameter H Earth equatorial radius Earth J2 3.9860 x 105 km³/s² To 6378.1 km J2 1.0826 x 10-3 (a) Simulate the orbit under the Earth J2 perturbation by using the following four different state rep- resentations: (i) Cartesian, (ii) Keplerian orbital elements, (iii) modified equinoctial orbital elements, and (iv) Milankovitch orbital elements, where use the initial condition given in Table 2 (defined in the ECI frame) and integrate the system for 15 orbits. For the simulation with orbital elements, use the Gauss planetary equations. Compare the numerical results of these simulations and verify the consistency. *NOTE: Perturbing acceleration must be expressed in the radial-transverse-normal (RTN) frame for Gauss planetary equations. (b) The Milankovitch element set has a constraint among its elements. Using the simulated result of the previous problem, evaluate the constraint over time and discuss.
(1.1) Perturbed two-body problem Consider Earth-orbiting spacecraft. Define Earth-centered Inertial (ECI) frame N: {0, 1, 2, 3} with its origin at the Earth center and having 3 aligned with the Earth rotation north pole. Let r = ïñ₁+y^2+zñ³ and v = v₁₁+v₁n₂+v₂n3 represent the position and velocity vectors of the spacecraft orbit, respectively. We consider the perturbation due to the Earth irregular gravity field (J2 only) acting on the orbit: a12 215 _342 [(1 [(1-5) Th xn₁ + 1+ (1-5) y yn2+3 +(3-5) zn] (1) where r = ||||2, and we take the Earth's equatorial radius as the normalizing radius, ro. Specific values for these parameters assumed for this problem is summarized in Table 1. Table 1: Assumed dynamical parameter values (Problem (1.1)) parameter Symbol Value Unit Earth gravitational parameter H Earth equatorial radius Earth J2 3.9860 x 105 km³/s² To 6378.1 km J2 1.0826 x 10-3 (a) Simulate the orbit under the Earth J2 perturbation by using the following four different state rep- resentations: (i) Cartesian, (ii) Keplerian orbital elements, (iii) modified equinoctial orbital elements, and (iv) Milankovitch orbital elements, where use the initial condition given in Table 2 (defined in the ECI frame) and integrate the system for 15 orbits. For the simulation with orbital elements, use the Gauss planetary equations. Compare the numerical results of these simulations and verify the consistency. *NOTE: Perturbing acceleration must be expressed in the radial-transverse-normal (RTN) frame for Gauss planetary equations. (b) The Milankovitch element set has a constraint among its elements. Using the simulated result of the previous problem, evaluate the constraint over time and discuss.
Elements Of Electromagnetics
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I am trying to code the solution to the problem in the image in MATLAB. I wanted to know what is the milankovich constraint equation that is talked about in part b.
![(1.1) Perturbed two-body problem
Consider Earth-orbiting spacecraft. Define Earth-centered Inertial (ECI) frame N: {0, 1, 2, 3} with its
origin at the Earth center and having 3 aligned with the Earth rotation north pole. Let r = ïñ₁+y^2+zñ³
and v = v₁₁+v₁n₂+v₂n3 represent the position and velocity vectors of the spacecraft orbit, respectively.
We consider the perturbation due to the Earth irregular gravity field (J2 only) acting on the orbit:
a12
215
_342 [(1
[(1-5) Th
xn₁ +
1+ (1-5) y
yn2+3
+(3-5) zn]
(1)
where r = ||||2, and we take the Earth's equatorial radius as the normalizing radius, ro. Specific values for
these parameters assumed for this problem is summarized in Table 1.
Table 1: Assumed dynamical parameter values (Problem (1.1))
parameter
Symbol Value
Unit
Earth gravitational parameter H
Earth equatorial radius
Earth J2
3.9860 x 105
km³/s²
To
6378.1
km
J2
1.0826 x 10-3
(a) Simulate the orbit under the Earth J2 perturbation by using the following four different state rep-
resentations: (i) Cartesian, (ii) Keplerian orbital elements, (iii) modified equinoctial orbital elements,
and (iv) Milankovitch orbital elements, where use the initial condition given in Table 2 (defined in
the ECI frame) and integrate the system for 15 orbits. For the simulation with orbital elements, use
the Gauss planetary equations. Compare the numerical results of these simulations and verify the
consistency.
*NOTE: Perturbing acceleration must be expressed in the radial-transverse-normal (RTN) frame for
Gauss planetary equations.
(b) The Milankovitch element set has a constraint among its elements. Using the simulated result of the
previous problem, evaluate the constraint over time and discuss.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fecc12e1f-7bf4-4124-a88c-68bd4f93278c%2Fyiz4zkf_processed.png&w=3840&q=75)
Transcribed Image Text:(1.1) Perturbed two-body problem
Consider Earth-orbiting spacecraft. Define Earth-centered Inertial (ECI) frame N: {0, 1, 2, 3} with its
origin at the Earth center and having 3 aligned with the Earth rotation north pole. Let r = ïñ₁+y^2+zñ³
and v = v₁₁+v₁n₂+v₂n3 represent the position and velocity vectors of the spacecraft orbit, respectively.
We consider the perturbation due to the Earth irregular gravity field (J2 only) acting on the orbit:
a12
215
_342 [(1
[(1-5) Th
xn₁ +
1+ (1-5) y
yn2+3
+(3-5) zn]
(1)
where r = ||||2, and we take the Earth's equatorial radius as the normalizing radius, ro. Specific values for
these parameters assumed for this problem is summarized in Table 1.
Table 1: Assumed dynamical parameter values (Problem (1.1))
parameter
Symbol Value
Unit
Earth gravitational parameter H
Earth equatorial radius
Earth J2
3.9860 x 105
km³/s²
To
6378.1
km
J2
1.0826 x 10-3
(a) Simulate the orbit under the Earth J2 perturbation by using the following four different state rep-
resentations: (i) Cartesian, (ii) Keplerian orbital elements, (iii) modified equinoctial orbital elements,
and (iv) Milankovitch orbital elements, where use the initial condition given in Table 2 (defined in
the ECI frame) and integrate the system for 15 orbits. For the simulation with orbital elements, use
the Gauss planetary equations. Compare the numerical results of these simulations and verify the
consistency.
*NOTE: Perturbing acceleration must be expressed in the radial-transverse-normal (RTN) frame for
Gauss planetary equations.
(b) The Milankovitch element set has a constraint among its elements. Using the simulated result of the
previous problem, evaluate the constraint over time and discuss.
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