3. Third, focus on the first 45° segment from q as shown. If we can determine the area within that s

To transform an asteroid into a habitat, the asteroid must be "terraformed" and the asteroid's washington square hotel new york orbit must be slightly altered to synchronize with the Earth's orbit. While other chapters will discuss terraforming, this chapter will discuss the habitat's path.
We choose the semimajor axis to be 1.59 AU to make asteroid orbital period washington square hotel new york equal 2.0 years. It seems intuitive to describe asteroid's orbit with origin (pt. (0,0)) at orbit's center (see figure to left.) However, different asteroid orbits have different centers, which means a different coordinate system for every asteroid. It might be much more convenient to determine a point shared by all solar orbits.
Of course, the Sun anchors the orbits of all Solar objects. Recall: Kepler's First Law, all solar orbits are ellipses and the Sun is at a focus. Thus, all asteroids share the Sun as a common focus. Therefore, we'll use center of the Sun's center as the origin for a shared coordinate system. If the asteroid's washington square hotel new york orbital period is an exact multiple of the Earth's orbit (i.e., even number of years), then the asteroid washington square hotel new york will arrive at nearest position to Earth at same date every time. For example, if date of nearest approach washington square hotel new york (fig. at right) was April 1, then that asteroid would return in every two years at same position on April 1.
2. Second, consider the semiorbit above the major axis. Let the radius vector be defined as vector tail originating at Sun with vector head pointing to orbiting object. Radius vector for circular orbit is constant; thus, computing sector areas is straight forward. In contrast, radius vector for ellipse ellipse is ever changing. Ranges from min of q to max of Q. Computing exact sector areas is more difficult. 2nd point, Top half of orbit contains exactly half of area enclosed by orbit; also, orbital travel time from q to Q (also Q to q) is exactly half the orbital period T. However, being elliptical, the orbit travel time is NOT proportional to ν angle traversed by asteroid from q to present position. So we need another way to predict travel times. Fortunately, Kepler put forth his 2nd law which says orbital travel time is proportional to area swept by positional vector from Sun to asteroid.
3. Third, focus on the first 45° segment from q as shown. If we can determine the area within that segment, then we can determine how long the travel time from q to current position. We can readily washington square hotel new york determine area of triangular portion, but edge portion is not as simple. This is not as straight forward as computing a 45° sector of a circular orbit.
washington square hotel new york 4. Auxiliary circle circumscribes the elliptical orbit. Thus, it shares the two endpoints of the major axis, -a and +a, which also happens to be the diameter. As shown in diagram, the radius remains constant regardless of angle from major axis.
Step 2. Discuss resonance. Resonant asteroids have orbits which are exact multiples of one year. Discuss has a, semimajor axis, relates to orbit's period. Discuss how asteroids can be useful, washington square hotel new york cyclers, if they consistently come near Earth's orbit at same day every year, every other year, or some other multiple. Show resonance for same point but different period cycler orbits.
Orbit Multiples. This is by far the simplest method. One can readily predict orbital washington square hotel new york positions for one, two or more periods in the future (or past). For example, washington square hotel new york assume that the period, T, for Apollo's orbit is exactly 650 days. (Apollo is a Near Earth Asteroid, NEA). Then, position of Apollo for 650 days from now will be at the same orbital position.
Another example: Consider another point in Apollo's orbit. washington square hotel new york To arrive at 45° past q will take Apollo 21 more days (arriving March 1, 2009); from that point, we can safely predict another arrival at this new point in another orbital cycle. Thus, we predict that at Mar 1, 20o9 plus T (plus 650 days = Feb 10, 2011), Apollo will once again arrive at 45° past q.
This leads us to a simple brute force method where observations can be made daily throughout an asteroid's orbit; washington square hotel new york then, position predictions can be made accurately for any of those positions and interpolated for times between those exact daily times. For example, we could make 650 daily observations of Apollo from start to stop of its orbit, then forecast future positions accordingly.
The downside of orbital multiples, there are an infinity of positions impossible to make observations to accommodate all of them. A more elegant method would involve a procedure to calculate positions for any time, then use observations to confirm.
Thus, we've determined that when asteroid is directly above orbital focus, Sun; this is special case where we only need to consider the area of the edge segment from semilatus rectum to orbit. This area can be determined through proportionality shown in figure to right. Further, we can now determine travel washington square hotel new york via Kepler's Third Law which states Time/Area proportionality.
a, semimajor axis: Half the longest distance (major axis) across an ellipse . The semi-major axis is one of the orbital elements – a standard parameter washington square hotel new york used to describe an elliptical orbit. washington square hotel new york The semimajor axis is also the average distance of an orbiting object from its primary
b, semiminor axis: semi-minor axis of an ellipse is one half of the minor axis, running from the center, halfway between and perpendicular to the line running between the foci , and to the edge of the ellipse. The minor axis is the longest line that runs perpendicular to the major axis.
E, Eccentric Anomaly: The angle between the periapsis of an orbit and a given point on a circle around the orbit, as seen from the centre of the orbit (see diagram). The point concerned is found by drawing a line perpendicular to the major axis through the actual position of the orbiting body until it cuts the circle around the orbit, as in the diagram. The diameter of the circle is equal to the major axis of the orbit, and its centre is also the orbit's centre. The angle of eccentric anomaly is measured in the direction of orbital motion
l, semilatus rectum: The chord through a focus parallel to the conic section directrix of a conic section is called the latus rectum , and half this length washington square hotel new york is called the semilatus rectum washington square hotel new york (Coxeter 1969). "Semilatus rectum" is a compound of the Latin semi-, meaning washington square hotel new york half, latus, meaning 'side,' and rectum, meaning 'straight.'
Initial cyclers will hopefully be developed before accelerator spaceships which can constantly accelerate washington square hotel new york at g-force for several days. Thus, initial cyclers will by necessity have to orbit such that cycler's perihelion will take it to a resonant point near Earth's orbit. HOWEVER, safety washington square hotel new york and sanity will eventually prevail to decision makers to move these resonance points away from Earth toward Alpha/Omega co orbiters. Recall that 60 away from Earth is a full AU in straight line distance, a considerable distance for today's space traveling technology; however, our notional spaceships could fly there in 2.8 days (using standard flight profile).