![]() ![]() It should be noted though that for purely circular orbits, there is technically no periapsis point, but you can still just arbitrarily define it and move on.Īs far as software for doing this, you can either write out the function yourself (which I would recommend doing at least once to practice it), or use already written programs. You can calculate the position and velocity vectors of the orbit with respect to the perifocal frame using true anomaly, a, e, and mu, and then once you do that you then use a rotation matrix to convert the coordinates of the position and velocity vectors from the perifocal frame to the ECI frame, and then you have your answer. Y-axis completes the right handed system (+z cross +x = +y) Z-axis points from the orbit center to the direction of the orbit's angular momentum (position cross velocity at any point in the orbit since its constant in a two-body orbit) X-axis points from the orbit center to the periapsis point in the orbit This orbit frame is known as the perifocal frame which is defined as follows: The 3 angles to describe the orientation of the orbital plane are the angles of a 3-1-3 euler angle sequence describing the orientation of the orbit frame with respect to the ECI frame. The semi major axis describes "the size", the eccentricity "the shape" and true anomaly tracks the position vs. The orientation of the orbital plane is described by the right ascension, inclination, and argument of periapsis. The process of converting between Keplerian / classical orbital elements to ECI state vector is outlined in a number of books, a popular one being "Orbital Mechanics for Engineering Students" by Howard D. Let me know if this explanation needs clarification. You now have all of the bits to create the X, Y, Z state from the data you were given. Finally, you know how long it has been since the spacecraft periapsis passage, so you can fully determine the true anomaly. Therefore, you can compute the orbital period from the mean motion and the GM of the Earth: the period is simply 2-pi divided by the mean motion. ![]() ![]() įurther, in a circular orbit, the angular rate is constant. So what's the starting vector which you need to multiply your direct cosine matrix with? Well, back-tracking to the start of the problem, we've stated that the spacecraft was described as rotating around your hand, and we've done all of the rotations with as the frame. This means that, after we've accounted for that rotation, we'll have done a 3-1-3 Euler rotation, where the angles are respectively the RAAN, inclination, and true anomaly. As discussed in the first paragraph, at this point, the motion of your spacecraft is fully described by a rotation of your thumb, i.e. The final rotation will be for the true anomaly of your orbit. So far, we've done two rotations: by the 3rd and the 1st axis. It corresponds to a rotation about your index finger, which is the 1st axis. Now, imagine what it means for your orbit to have an inclination with respect to your right-hand. That corresponds to an Euler rotation by the 3rd axis. Therefore, in terms of rotation, the RAAN corresponds to a rotation about the a-cross-b vector (i.e. The RAAN is measured from the origin frame, therefore from the a vector. Looking at the right-hand rule diagram below, that orbital plane is the plane created from the vectors a and b. What does it mean in terms of geometry? It means that spacecraft will always cross the orbital plane at that angle compared to the "origin" of the frame. Think of the right ascension of the ascending node. Looking at the right-hand rule diagram below, it means that the motion of your spacecraft corresponds only to a rotation around your thumb. You know that you've "reached" the destination frame when the motion of the spacecraft can be represented by a rotation around a single axis. We'll work through finding the correct "destination" frame. Specifically, its furthest distance from the origin is the semi-major axis, but since we're in a circular orbit, the spacecraft's only distance is the sma. Imagine that your right-hand is centered on the Earth, so a spacecraft will be in orbit around your hand. ![]() Using Ampere's hand / right-hand rule, "start" with your hand aligned with the "origin" ECI frame. Start by thinking of it in terms of Euler rotations. ![]()
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