As discussed previously, our goal is to compute the TA area to help determine orbital flight times;

Select from following: TRUE ANAMOLY: shows orbital position with respect to perihelion. luxury hotels florida AUXILIARY CIRCLE: circumscribes elliptically shaped orbit. luxury hotels florida ECCENTRIC ANOMALY: maps asteroidal position from elliptical orbit to the auxiliary circle. MEAN ANOMALY (1ST MEANING): redefines TA area in a meaningful way. MEAN MOTION: asteroid's average motion luxury hotels florida throughout the orbit. MEAN ANOMALY (2ND MEANING): redefines duration due to orbital mean motion. luxury hotels florida KEPLER'S EQUATION relates the two meanings of MEAN ANOMALY. SEMILATUS RECTUM
luxury hotels florida Orbital positions can be expressed as anomalies (or differences) from a reference point (usually the perihelion, orbit's nearest point to the Sun). These anamolies are usually expressed as angles (degrees or radians).
Predicting orbital positions in a circular orbit is straightforward because objects in circular orbits move at a constant luxury hotels florida rate; thus, extrapolating duration to future positions is a simple matter of proportionality.
Predicting positions for an elliptical orbit is slightly more complicated because these in-orbit velocities vary between a range of values. Kepler's equation presents an elegant, general solution for predicting orbital positions in elliptical orbits.
This angle is the "True Anomaly" or ν "nu". Since the reference ray (Sun to the perihelion) is 0°, we're primarily interested in the S-y e ray, the positional vector, R, from Sun to object's position in the orbit.
Thus, for same duration, asteroid travels greater luxury hotels florida orbital distances when closer to the Sun and a lesser luxury hotels florida distance when further away. This accords with Isaac Newton's work in gravity which proved that Sun's closer satellites have greater speeds then those further away.
Eccentric anamoly (EA) area can be easily calculated because it is a simple circular luxury hotels florida sector. luxury hotels florida EA's apex, C, is at orbit's center, and side opposite C is the border of auxiliary circle (circular segment, q to y c ).
For every EA, there is a corresponding TA with area as shown. True Anomaly area (A ν ) is determined by three points: y e -S-q. A ν is not so easily determined as A E . While A E is bounded by a circle, A ν is bounded by an eclipse. Also, TA's angle apex, S, is not in the center of the orbit.
As discussed previously, our goal is to compute the TA area to help determine orbital flight times; unfortunately, TA's shape complicates matters. TA's outer edge (orbit) is ecliptic, and TA apex is not at orbit's center.
Can we easily modify EA to readily determine TA area?? Towards that end, we hypothesize there exists a circular sector with area equal to that of the TA area; we'll call this sector's angle, the Mean Anomaly.
EA area could be readily approximated via inscribed luxury hotels florida triangle. By inspection, we see triangle, luxury hotels florida q-C-y c , contains all EA area except for a small portion of circle between the chord, q-y c , and corresponding arc, q-y c .
Like the EA, TA area can also be approximated by an inscribed triangle. luxury hotels florida Triangle, q-S-y e , contains all TA area ecliptic portion except for a small portion of eclipse between the chord, q-y e , and corresponding ecliptic arc, q-y e .
Mean motion, n, is the average luxury hotels florida angular velocity of a satellite progressing through one cycle of an orbit . For circular orbits, the angular velocity is constant; thus, the mean motion will also be the actual angular velocity. However, most orbits are elliptical with angular velocities which vary according to the instantaneous position vector. Thus, mean motion is only an average value which will coincide with a very few midrange angular velocity values experienced during an orbit.
Traditional luxury hotels florida Method T = 2π√(a 3 /μ)= 2π a √(a/μ) Traditional values gives orbital period in seconds. T = 2π * 2.205 x 10 11 m * √[(2.205 x 10 11 m)/(13.27 x 10 19 m 3 / sec 2 )] T = 13.85 x 10 11 * √(.1661 x 10 -8 sec 2 ) T = 5.645 x 10 7 sec
For either mean motion or orbital luxury hotels florida period, we note the only variable luxury hotels florida is the semimajor axis, a. Furthermore, if we squared both sides, T squared would equal a cubed times some constant luxury hotels florida ( T 2 = k a 3 ). This complies with Kepler's 3rd Law.
Mean Anomaly, M, has a second meaning; it relates mean motion to the auxiliary circle. Consider a notional object traveling on the auxiliary circle. Furthermore, let this object's velocity equal mean motion which was previoualy discussed. In that case, Mean Anomaly luxury hotels florida measures the angular distance traveled for a given time duration, t.
Mean Anomaly's shape continues to be same for first and second meanings. Mean Anomaly's Apex, M, is at orbit's center, and reference ray is from M to q, real orbit's perihelion. Outer edge of sector is the arc from q point q to notional object's present position.
The semi-latus rectum ( l ), of an ellipse is the distance from a focus to the ellipse itself, measured along a line perpendicular to the major axis. Thus, if x-value of asteroid's position happens to coincide with orbit's focus (we'll call S which could be Sun), then the segment S-y e represents the semilatus rectum.
However, we're primarily interested in relationship between True Anomaly, ν and elapsed time (duration since asteroid last passed perihelion, closed point of orbit to Sun). Therefore, following table dispenses with E.