Polynomial Fits For NEO Observations

 

  Trying to determine the relative position and velocity for a NEO can be difficult if there is curvature present in the object's 3D track. The fit may need to be extended to higher degree polynomials. The polynomial is a function of time and since we are working in 3D the coefficients will be vectors. The computed position for the object has to be the same as the observer's position in three dimensions plus a distance λ_k along the observer's line of sight. Using the Method of Least Squares from the Calculus of Variations one can set the deviation for the observation equal to the difference between the two expressions for the position. It is a vector and the variance which is the sum of the magnitude of each deviation is minimized. For the best fit the change in the variance is zero for a change in any of the coefficients. This assumption allows us to derive a set of normal equations whose solutions will give us the coefficients. An expression for the unknown distance along the line of sight can be found by solving the equalent expressions for the position for each observation separately.

 

 

  The normal equations are similar to those for polynomial fits used for simple linear regression with an additional projection operator in the sums that is a result of only using sighting directions and not distances. Since the coefficients are vectors the normal equations that result involve the sum of the sum of the product of a set of matrices and the coefficients. In the sum p ranges from 0 to the highest degree of the polynomial. For a quadratic fit C₀ would be the initial position of the object, C₁ would be the velocity and C₂ would be half the acceleration. The normal equations can be solved by merging the separate matrices into a single matrix and the vectors into a single vector and the resulting equation can be solved in the usual way. The coefficients can be separated afterwards. More accurate observations and a greater number of observations will give a better fit.

 

  Supplemental (Jan 29): Virtual displacements are used in Lagrangian mechanics and the Calculus of Variations but minima problems are much older. Hero of Alexandria proved that light reflecting from a mirror takes a minimum path.

 

More Moderates Needed In Congress?

 

  The vote in the November elections has shifted a few seats in favor of the Democrats in the 113th Congress but not enough to get Congress out of its doldrums. A quorum is needed to get anything done and with the two parties so nearly evenly balanced it makes it difficult for either side to get one together. Any internal opposition within a party can effectively block the party's proposed legislation. The two Congressional chambers are divided by party leadership and anything passed by either body is even less likely to become law. Voting for hardliners this time may have been a mistake since it's more likely to result in a disfunctional Congress. It presents an opportunity for Moderates to work to bridge the gap and help put together a quorum to pass needed legislation. It also shows that voting for third party candidates is not a wasted vote. If a moderate third party were able to deny both parties a quorum and side with either one on a vote then it could shift the balance of power in the direction of its political platform and justify its existence.

 

Two NASA Climate Change Videos

 

src: NASAexplorer

 

src: NASAexplorer

 

  NASA recently posted two videos on YouTube dealing with climate change. The first shows the statistical distribution of summer temperature anomalies for the Northern Hemisphere by decade from 1950 to 2001. The distribution is relatively stable for the decades whose centers range from 1955 to 1980 then the mean temperature of the distribution starts to increase. The temperature anomaly is difference between the measured temperature and a reference temperature. The trend has been one of an increasing anomaly over the last century. What happened about 1980 that might have caused the increase? The Three Mile Island accident occurred in 1979 and the number of nuclear reactors that have been licensed since then has fallen off. The anomaly was relatively constant over the period from the Forties to Seventies when they were constructing reactors. Could turning away from nuclear energy have caused global warming? We have had to rely more on fossil fuel plants which released CO₂ into the atmosphere. But fossil fuel consumption shows a steady increase from 1950. The record for global biomass burning shows some correlation with the temperature anomaly but not during the last 50 years. Mt St Helens erupted in 1980 but there have been other large eruptions too. It is difficult to link cause with effect when there are so many possible contributing factors. We need to understand what is happening before we can choose the actions that will help remedy the situation.

 

  Supplemental (Jan 19): The amount of ice in the world is a remnant of the last ice age and it has been slowly disappearing over the centuries. When the world loses heat during winter it has a choice of cooling ice or making more ice. The formation of ice acts as a floor for low temperatures and transports the cold to cool the summers. Water evaporating in summer also helps cool the land but less melting snow and ice leads to warmer and dryer summers. This may be what we are seeing in the temperature anomaly. One might make an analogy of a plane on a runway gaining speed. The faster it goes the more lift it gets but it needs to reach a certain speed before it takes off. This speed may be the tipping point that the environmentalists are like to refer to. We may be close to the turning point but don't have enough speed yet to get airborne. Some parts of the world are already airborne. They don't have snow at any time of the year. Other areas are grounded all year and there is no time that they are free of snow or ice. The world is in transition and has been for a long time now with the boundaries moving northward.

 

The MSL Jan 15th Teleconference

 

src: jplnasa

 

  Tuesday's JPL teleconference dealt with some of Curiosity's findings in the Yellowknife area over the last few weeks. The two most interesting items were some vein-filled material and some cross layering deposits. Chemical analysis indicates that the vein-filled material is most likely hydrated calcium sulfate that appears to have been left by water seeping through cracks in the rocks. The cross layering deposits with a variety of slopes are indicative of periodic water flows across the surface. Another discovery is a layer containing spherical objects which have been previously described as being formed in a water environment. Curiosity's plans include spending a few more days at the present location before it moves on to a nearby location dubbed John Klein where it will drill into the rocks there. They are also hoping to get a better view of the vein-filled material by breaking the rocks along a crack. After this Curiosity will move on to the lower reaches of Mt Sharp. telecon images & associated press release

 

The Advantages of a Stable Reference Point

 

  All distance observations are relative measurements. We have an object and an observer. We can't discuss the motion of an object by itself so we can never remove the observer from the observation completely. When we have a group of observations there is an advantage to having a common point of reference to which individual observations can be referred to. For a system of bound particles all the particles move relative to a center of mass and with it. The center of mass has only has the motion of the group and no individual motion so it is the most stable point for the group. The rotational motion of the Earth is relative to its center of mass and that of the Earth-Moon system is relative to its barycenter. This is why the two points are the natural choices for common reference points. They are points of relative stability. Observations from these points tend to be more consistent than those of individual observers. They simplify the relative motion of an object. Neither point is without some acceleration of its own so we can't completely remove the motion of the reference point from the object's motion but we have minimized it. Over short periods of time the acceleration of an object may have a negligible effect and we can make a better linear least squares fit of the position and velocity. When we notice some curvature in the path of the object it doesn't matter whether it is acceleration relative to a Galilean reference frame or acceleration relative to the reference point. A higher order polynomial may be needed to fit the data but we should be able to do this since the polynomial is linear in terms of its coefficients. A linear fit would be the minimum necessary to determine an orbit since the position and velocity are all that is needed to calculate the orbit's specific energy and specific angular momentum and consequentially the Keplerian orbital elements. Perturbing bodies like the Earth and the other objects in the solar system can cause an object's motion through the solar system to deviate from an ideal Keplerian orbit but we still have enough information to calculate an approximate path. Observational errors make the path somewhat uncertain but small deviations to the position and velocity can be used to compute alternative paths.

 

The Effect of Observer Position on Tracking

 

 To see how the observer's position affects tracking one could look at one's thumb an arm's length in front of one's eyes and look at the thumb one eye at a time. The thumb will move sideways relative to its background. Moving one's head back and forth sideways while looking through one eye will also affect the thumb position relative to the background but even though the head is moved it doesn't significantly affect the angular change that result from switching eyes. This is because the distance of the thumb hasn't changed and the range parallax depends of the relative positions of the eyes so the effect of the head's motion is canceled out. Motion of the head sideways will still affect the position of the thumb on the background and as a consequence its direction relative to the head.

 

  Switching eyes corresponds to the simultaneous sighting method used to determine the object's position. The motion of the Earth doesn't affect range measurements relative to its center. Changes in bearing due to the Earth's motion are part of the object's relative motion. The Earth and the Moon orbit a common center known as the Earth-Moon barycenter. As a result of the Earth's rotational motion an observer's position is periodically altered by the Earth's diameter in half a day. The Earth's motion relative to the Earth-Moon barycenter shifts the Earth's position by roughly a Earth diameter in half a month. The result is that range measurements are more likely to be affected by the least squares method than paired observation parallax method. Making observations relative to the Earth-Moon barycenter would result in a better agreement of the object positions from the two methods. We would prefer to remove the way an observer affects the observations as much as possible. The Earth-Moon barycenter would be the better choice for comparison of observations that span a month of time. The perspective could be changed after the least squares fit is done.

 

  The precision needed to make predictions of where an object will be from one night to the next based on tracking information is not as great as that needed to do orbit calculations. Care needs to be taken to ensure that a common perspective is being used when when working with sets of observations.

 

Least Squares NEO Tracking Fits

 

  The simple NEO tracking problem that we did for two observers in Sydney and Perth required simultaneous sightings on an object. Using the method of Least Squares we can combine a number of sightings taken at different times to get another estimate of linear 3D motion. The following example uses the same sightings as before but computes positions in the fixed celestial frame by rotating the geographic coordinates of the observers by the amount indicated by the GMST for the time of the sightings.

 

 

 

 

  We get a fit that is very close to the results of the previous calculations. Note that the unit of time here is the day so V₀ needs to be divided by 24 to get the speed in R_E/hour.

 

NEO Tracking Calculations in the Celestial Reference Frame

 

  I redid the previous somewhat simplified NEO tracking problem using celestial coordinates for the sightings and geographic coordinates for the observer locations. For each pair of sightings one needs to find the observers' positions in the celestial frame by determining the Greenwich Mean Sidereal Time (GMST) for Earth's orientation at the given time and then rotate the position coordinates. One also has to convert the Right Ascension and Declination of the observations into celestial unit vectors for the parallax calculation to work properly. In the Mathcad function shown below the procedure goes as follows. First calculate the GMST using the Julian Date and convert hours to an angle. Then create a matrix to do the rotation about the Earth's celestial pole and use it to rotate the observer positions. Finally one calculates the distance along the line of sight to determine the position of the object.

 

 

  Suppose that on Jan 12, 2013 two pairs of sighting on an object are made from Sydney and Perth at 12:00 UT and 13:00 UT. The coordinates for positions and sightings are as indicated. The procedure gives the object's position in the celestial coordinate frame and one can convert the Cartesian coordinates to celestial angles and the range of the object. As was done previously, the RA is in hours and the range is given in Earth radii.

 

 

 The second sighting is done the same way and one again gets a speed of 0.8 R_E/hour for the object.

 

 

  If one needs more accuracy the JDPe2X procedure could be modified to take into account precession, nutation, the motion of the Earth about the Earth-Moon barycenter, etc. in order to get a better position for the observers. The rotation for the first two corrections can be found by accurately measuring the position of the Earth's celestial pole and checking the sidereal time for the Greenwich Meridian. By definition the Equinoctial point is determined by the intersection of the Equatorial and Ecliptic planes.

 

  Supplemental (Jan 13): The Earth-Moon barycenter is closer to a Galilean reference frame than the geocentric frame but since the intended goal is to track the NEO's motion relative to the Earth the geocentric frame will suffice. The position of the observers relative to the Earth's center is all that is necessary to measure the parallax and calculate the object's geocentric range.

 

Sidereal Time

 

  If a number of observers at different locations around the world want to track an NEO in real time relative to the background stars they need to adopt a common reference frame. J2000 geocentric equatorial coordinates would be a convenient choice since the positions of the stars in this frame are known. The positions of the observers need to be adjusted as a consequence of the Earth's rotation since they will affect the apparent position of the NEO due to its parallax.

 

 The Earth's rotation about its axis determines a specific direction in space, the celestial pole, from which we get the equatorial plane. We need another point in space to act as a reference point for rotation and the place where the Sun crosses the Equator, the direction of the Equinox, is the customary direction. This allows us to use Right Ascension (RA) and Declination (Decl) as our astronomical coordiante system. The subsolar point on the Earth's surface at the time of the Equinox points to the hour angle of RA as the Earth rotates. This hour angle is used to define the mean sidereal time for the Greenwich meridian which changes at a steady rate as the Earth rotates. On January 1, 2000 12:00 (J2000.0) the Greenwich mean sidereal time (GMST) was 18h 41m 50.548s. The rotational period of the Earth relative to the stars is 23h 56m 4.091s. By just using these two facts we can get a fairly good estimate of the sidereal time for Greenwich. For other locations we have to convert the longitude of the location to hours by dividing by 15deg/hr and then add the result to the GMST to get the local sidereal time (LST). Some Mathcad functions that do this are shown in the image below. The Julian date (JDate) is the number of days and a fraction from the arbitrary beginning of the period. The astronomical day started noon which was more convenient for observations during the night. Astronomers now count years in Julian years of 365.25 days which is why J2000.0 converts to an whole number. The Julian date was used to measure the elapsed time in days.

 

 

  This will work fairly well for short periods of time. One will find more complicated formulas for the sidereal time but these take into account the change in the position of the Vernal Equinox as a consequence of the Earth's precession and nutation. The shape of the Earth and the increase in the distance of the Moon from the Earth can also alter the Earth's rotational rate. For the short period of time that an NEO is near the Earth as it is being tracked these changes can be neglected if one is only interested in the NEO's relative motion.

 

  Supplemental (Jan 13): The function for the GMST could be modified so that the result falls between 0h and 24h as was done for the LST. Both functions were originally parts of a single function for the LST that was split and I neglected to reduce the result for GMST.

 

Tracking an Object in 3D

 

  Suppose that the two observers in the last blog take a second sighting on the 1° parallax object an hour after the first and calculate a new position for it. (To simplify the problem the Earth's rotation has been neglected.)

 

 

  The new position can be used to determine how much the object has moved relative to the Earth in the elapsed time. They will find that it has moved 0.8 Earth radii so its speed relative to the Earth will be 0.80 R_E/hour.

 

 

 Once our observers have the geocentric position and velocity relative to the Earth they can make predictions about where it will be at future times and will be in a better position to track the object as it moves across the night sky. Taking into account the Earth's motion relative to the Sun the data from the observations could also be used to determine object's motion under the influence of the local gravitational field. When sighting a distant NEO our two observers could make two sets of observations exactly one Earth rotation apart so that the calculations can be done in the Earth's reference frame. A more practical approach would be to do everything in the geocentric celestial reference frame and adjust the observers positions for the Earth's rotation since photographis observations are done relative to the background stars.

 

Calculating an Astronomical Object's Parallax and Geocentric Position

 

  Astronomers have been measuring the parallax of astronomical bodies since at least the time of Ptolemy and Hipparchus. Ptolemy calculated a parallax for a position of the Moon and got a value of 1+7/60 degrees. The Sun had no measurable parallax. Unfortunately, like meteorites, comets were considered phenomena of the ethereal realm so we have no information about the path of comets from him. Tycho Brahe observed comets by measuring their positions from nearby stars to about 1 minute of arc accuracy.

 

  When tracking an object passing near the Earth one observer can only determine its direction at various times if one does not take advantage of a theory of motion. To get the range of the object one must make simultaneous observations from two places on the Earth's surface. Consider the situation shown in the figure below where two observers sight object X from positions P₁ and P₂. The distances of the object from the two positions will be λ₁ and λ₂ along directions e₁ and e₂ respectively. It can be shown that the two lambdas are solutions of the set of equations below where ΔP=P₂-P₁ and the dot indicates a vector dot product.

 

 

 Now consider two observers, one in Sydney and a second in Perth, who can both sight an object whose parallax is 0.1° in the directions shown. We can solve the equations above to determine the geocentric distance and range of the object. For simplicity it can be assumed that the radius of the Earth is 1 and we see that our observers will be separated by about half an Earth radius. The range of the object is found to be 294 Earth radii.

 

 

  If the parallax is increased to 1° the range of the object decreases to 30.2 Earth radii.

 

 

  For a object at a distance of about 5 Earth radii the parallax for observers separated by 1° is about 0.2°.

 

  Edit (Jan 9): Corrected e₂ so that the parallax worked out correctly but with altered values for the corresponding ranges.

 

Amateur Observations of Comets and Asteroids

 

  When an amateur astronomer observes an object such as a comet or asteroid passing near the Earth the only data that he usually can get are a number positions in the night sky and their times. The positions can be found by triangulating the object's observed position from the known positions of nearby stars. If one plots the data on a celestial sphere one will find that the object tends to follow a great circle on the celestial sphere. To illustrate this I plotted the binocular observations of the positions of Comet Hyakutake that I made in 1996 on a sphere.

 

 

  These observations were made shortly after sunset and shortly before sunrise as weather permitted from Mar 21 to Apr 23 of 1996. Something similar could be done with 2012 DA14. But now one can document the observations by astrophotography with a digital camera.

 

2012 DA14's Plane of Relative Motion

 

  If one does a least squares fit for 2012 DA14's plane of motion on the geocentric celestial sphere one will find that the normal to the plane is near the direction of RA 5.749 hr and Decl 3.250°. In this plane the asteroid does a flyby of the Earth and is deflected by an angle of 28.0° as a result of passing through the Earth's gravitational field.

 

 

 The width of the plot is 60 Earth radii which is approximately the average distance of the Moon from the Earth. One can see how much closer the asteroid comes to the Earth than the Moon. The orbit of the asteroid will change in a manner similar to that in which path of a spacecraft is altered by a gravity assist maneuver. The asteroid is 45 meters across which is about half the length of an American football field.

 

  The JPL ephemeris uses 188 observations from about a year ago to compute the path the asteriod will follow. The computed positions were geocentric. An observer on the Earth's surface may notice some paralax as a result of the asteroid's proximity. This paralax could be used by a group of amateur astronomers to determine the distance and path that the asteroid takes as it flies by the Earth next month.

 

2012 DA14 Fly By on Feb 15, 2013

 

  A near Earth asteroid, 2012 DA14, will pass within 35,000 kilometers of the Earth on Feb 15, 2013 at 19:25 UT according to JPL's ephemeris. If the direction of the asteroid as it flies by the Earth is plotted in rectangular coordinates one gets an odd curve indicating that it will spend most of its time near the poles.

 

 

  The path appears to be a little simpler if it is plotted on a celestial globe and is very nearly a great circle. The swing from the Celestial South Pole to the Celestial North Pole takes place at the time of closest approach. The positions are given for one hour intervals. The center of the Celestial Sphere is 12h RA. The relatively large south to north motion is due the the fact that the asteroid is in an orbit similar to that of the Earth's and travels along with it.

 

 

  The path appears to wrap around the Earth making an angle greater than 180°. The asteroid will be deflected by its proximity to the Earth but as it moves farther away the angle relative to the Ecliptic will also decrease since the relative distance above orbital plane changes at a slower rate. The rate of change for the distance from the Earth is fairly linear and appears to be "V" shaped over the 20 days included in the plot. The bottom of the "V" is rounded at the time of minimum distance. The apparent magnitude will be greatest near close approach rising to 7.48.

 

  Supplemental (Jan 4): The JPL ephemeris was set to generate geocentric positions which are relative to the Earth's center and so the minimum distance from the Earth's surface will be about 29,000 km or 4.5 Earth radii. For comparison the average distance to the Moon from the center of the Earth is about 60 Earth radii.