Figure 1. Earth, Orbit and Sun Geometry with local noon occurring at maximum latitudes
1. INTRODUCTION
This text explains how the local mean solar time is estimated for the TRMM overflights, and explains why there is a systematic pattern in the time of day that overflights are encountered.
The local Mean Solar Time (MST) provides a convenient measure of diurnal phasing of the data collection, but is only a rough measure because it doesn't account for seasonal variations in Local Solar Time (the equation-of-time corrections). It shouldn't be confused with Local Time in specific time zones, since it uses a simple first order time-of-day correction as a function of longitude around the Earth.
For the TRMM Overflight Finder tool, the Universal Time of each overflight is estimated as follows: 1) The time of the orbit start is taken from a database or estimated by a linear fit for future orbits. 2) The time to the overflight in that orbit is estimated using a polynomial fit of time vs. the longitude of the overflight relative to the orbit start. At the prime meridian, at Greenwhich England, the UT is the same as the local time and local mean solar time. To get the local Mean Solar Time from UT for other locations, we simply add 1 hour for each 15 degrees east of the prime meridian.
The reason that the local-time-of-day-of-overflights at any location change gradually on about a 47 day cycle, has to do with the TRMM orbit precession relative to the Sun direction. The geometry of this precession is explained below.
2. ORBIT GEOMETRY "SNAPSHOT" FOR ONE DAY
A key item to keep in mind to understand the data collection times is that the Earth rotates relatively rapidly relative to the orbit plane and sun position. We will discuss the rate of change of the orbit plane in the next section, but for the purpose of data collection over 1 day, we can consider the orbit relatively fixed.
For example, consider a situation with the Sun in the Earth equatorial plane, (equinox conditions) and the 35 degree inclination orbit reaching its maximum latitude in the direction along the sun line. This is illustrated in Figure 1.
Figure 1. Earth, Orbit and Sun Geometry with local noon occurring at maximum latitudes
Note, North is up, and the daylit side of the Earth is on the right. The orbit track has the ascending path near the viewer, since the orbit sense is generally aligned with Earth rotation direction for low inclination orbits. The Earth rotates positively in the right hand sense about the North pole, counterclockwise as seen from the North.
The Earth rotates under the orbit plane one full rotation every 24 hours, while the orbit plane stays nearly fixed relative to the sun. There are about 16 orbits in each day. One can see that for the geometry of Figure 1, each time the maximum latitude (35 degrees) is reached in each orbit, the local time of day for sites under the satellite at this latitude is near noon. Likewise one can see that each time the minumum latitude (-35 degrees) is reached, the local time of day for sites under the satellite there is always near midnight. At the equator, the ascending path of the orbit always is collecting data around dawn, and the descending path of the orbit is always collecting data around dusk, with the sun in this particular orientation to the orbit plane.
Thus by visualizing the orbit plane on a selected day, one can see that various places around the orbit each collect data at certain local times. The Earth rotation brings various specific sites under the satellite which are spread out around the globe, about 16 per day, but for each latitude, the local times--and the sun geometry at that latitude--is about the same.
As another example, see Figure 2 which shows a situation with the 35 degree inclination orbit reaching its maximum latitude on the night side of the Earth, near midnight in the direction opposite the sun line. Maximum latitudes now collect data around local midnight, while the northbound equator crossings get dusk data and the southbound equator crossings collect dawn data. Due to the precession of the orbit plane, as discussed further below, the geometry in Figure 2 will occur roughly 23 days after the geometry in Figure 1 occurs during the TRMM mission.
Figure 2. Earth, Orbit and Sun Geometry with local noon occurring at minimum latitudes.
Note, North is up, and the daylit side of the Earth is on the right. The orbit track now has the descending (southbound) path near the viewer. This is the geometry which will occur roughly 23 days before or after the Figure 1 geometry.
3. ORBIT PRECESSION RATE AND THE 46 DAY CYCLE
The data collect times evolve because of how the orbit plane gradually moves relative to the Sun. The orbit precesses about the North pole in a retrograde sense, so that the precession direction is opposite the Earth spin axis (North pole). The cause of this precession is discussed briefly below in Section 5, but the key point is to understand how this orbit geometry gradually changes and affects the local times of day of overflights. Since the precession rate relative to the Sun is about 7.765 degrees per day, the orbit plane orientation relative to the sun changes on a 46 day cycle. One can visualize the orbit plane as shown in figure 1 rotating about the South pole due to precession. While 35 north overpasses are at about local noon for the geometry snapshot in Figure 1, a few days later the 35 North passes would be moving earlier in time over the morning quadrant of the Earth.
More details of the orbit precession and rate estimates are provided in Section 5 below.
4. DATA COLLECTION PATTERNS AT VARIOUS LATITUDES
Figure 3 shows a sample plot for 1998 for overflight times over 35 degrees north, 0 degrees longitude. This shows a sawtooth like function as the time of day of the overflights cycle through the day every 46 days. Longitude of the site does not matter for the overall nature of this plot, which we confirmed with various samples using the overflight finder Local Sampling Times Graph option. The local time phasing (the approximate local time of data collection vs day of the mission) is independant of longitude, and only depends on latitude.
Figure 3. Local Times of Overflights within 50 km of 35 North (0 East) for 1998.
Figure 4 shows 60 days at 35 degrees north at the beginning of June 1998. This more clearly illustrates the time sampling pattern during a typical 46 day cycle.
Figure 4. Local Times of Overflights within 50 km of 35 North (0 East) for 60 days starting June 1 1998.
Figure 5 shows 60 days at 20 degrees north. This now shows 2 separate "lines" in the samples (which are plotted with the + symbol to show the individual overflights. The search radius/swath width criteria was set to within 150 kilometers from the specific site (20 north, 0 E/W) to pick up an adequate number of overflights. Further expanding the swath width will pick up more samples along the lines. The two lines correspond to times caught on the ascending and descending passes of the site.
Figure 5. Local Times of Overflights within 150 km of 20 North (0 East) for 60 days starting June 1 1998.
The separation of the pass times into two "lines" and the separation of these lines in time, as a function of latitude is discussed more in Section 7.
Figure 6 shows 100 days at 0 degrees north, on the equator. There is still a pair of lines here corresponding to the ascending and descending passes, but they are now evenly spaced because these passes are about 12 hours apart at the equator. The swath width criteria to within 300 kilometers from the specific site (0 north, 0 E/W) to show a good number of samples in each line. The pattern within the lines has to do with passes being caught to the east or west of the site as the days progress. The pattern is of the same nature, just different specific points for different longitudes.
Figure 6. Local Times of Overflights within 300 km of 0 North (0 East) for 60 days starting June 1 1998.
Note that the detailed structure of the sampling times may not be highly accurately depicted due to uncertainties and simplifications in our algorithm for selecting the local times of overflight. Detailed analysis of these uncertainties has not been completed as of this writing, but it is noted that the overflight local mean solar times may only accurate to about 10 or 20 minutes. Nevertheless the global trend in sampling times, accurate to the hour certainly, is illustrated.
5. TRMM ORBIT PRECESSION AND SUN GEOMETRY RATES
Low earth orbits precess (in the sense of rotational dynamics) because the Earth's oblateness applies a torque perpendicular to the angular momentum of the orbit. The torque can be thought of as trying to move the orbit plane toward the equator. Without this torque (or other disturbances) the orbit would stay fixed in inertial space, and cross the ascending node (the northbound equator crossing) at the same Right Ascension every orbit. (Right Ascension is the astronomer's coordinates for "longitude" in the inertial coordinate frame of the "fixed" stars.) The reaction to this torque however causes the Right Ascention of Ascending Node to change slowly. Since most spacecraft orbits have their angular momentum axis in a Northern direction (to take advantage of the Earth's rotational velocity in achieving orbit), the typical reaction is a slow decrease in the Right Ascension of the Node. (Right Ascension increases positively in the sense that the Earth Rotates.) A formula given in Wertz (Spacecraft Attitude Determination and Control, D. Reidel Publishing, page 68, formula 3-41d) gives the rate of change of the Right Ascension of the Ascending Node of the orbit (RAnodeRate) as a function of the orbit Semi-Major Axis (a), orbit eccentricity (e) and orbit inclination (i) as
RAnodeRate = (-2.06474 * 10^^14) * (a^^(-7/2)) * cos(i) / (1-e^^2)^^2
For the initial TRMM operational orbit parameters, a = 6728.14 km , i = 35.0 degrees, and e = 0.0, we get a node rate of -6.78 degrees per day.
RAnodeRade = -6.78 deg./day (for TRMM at a 350 km altitude)
This has an uncertainty of about .005 based on the variability of the semi-major axis , a = 6728.14 +/- 1.25 km for TRMM, which I expect is the dominant effect of concern for TRMM. The actual node rate could be refined based on flight data, but this number will be quite adequate for predicting the node for months in advance.
The Right Ascension of the Sun changes by 360 degrees in about 365.256 days (from the full orbit of the Earth around the Sun in one year). Thus we have a mean Sun rate of
SunRate = +0.98561 degrees per day
Due to the eccentricity of the Earth's Orbit, the actual rate varies somewhat, but this mean rate will be fine for our calculations. (The Sun gets about 20 minutes ahead or behind the mean rate, in a well studied manner known as the "equation of time.") Thus the total motion of the Sun Right Ascension relative to the orbit plane Right Ascension adds up to about:
SunRate_wrt_node = 7.765 deg./day
This means that the Sun position relative to the orbit plane changes on a cycle with a period of about 46.36 days. A rough uncertainty estimate on this period, based only on our node rate uncertainty, is just about +/- .03 days. I am not sure if other orbit effects could have a bigger impact on this period but I doubt it. This period agrees with estimates from a long term "solar beta" prediction for TRMM. This is the period of the cycle for the "Solar Beta Angle" which is a key driver for the times of day of site overpasses.
6. SOLAR BETA ANGLE DEFINITION AND EXTREME VALUES FOR TRMM.
We define the Solar Beta Angle as the elevation of the Sun in the orbit plane. A zero Beta Angle indicates the Sun is in the orbit plane. Positive angles are taken toward positive orbit normal, which is the spin vector of the orbit in a right hand sense.
While the phase of the Sun in the orbit plane changes 360 degrees each orbit, the Solar Beta Angle changes only slowly as the orbit precesses. The range of values found for the TRMM Solar Beta Angle is about +/- 58 degrees, which is the sum of two terms:
The way these combine can be thought of as follows: As the orbit precesses a maximum Beta angle is reached when the orbit plane is tilted most away from the Sun direction. This orbit tilt is always 35 degrees from the Earth's equatorial plane, and depending on the time of year, the Sun may be +/- 23 degrees from the Earth's equatorial plane. Depending how these combine, the maximum solar beta angle during one cycle of orbit precession can be between 12 and 58 degrees. Likewise the minimum solar beta angle during a cycle of orbit precession can be between -12 and -58 degrees.
The solar beta angle follows a 46 day cycle (at +/- 35 degrees amplitude) due to the orbit precession relative to the Sun, and an annual cycle (at +/- 23(.5) degrees ampitude) due to the Sun's path relative to the Earth's equatorial plane (the ecliptic plane in Astronomy, or the analemma path projected on the Earth).
The important point for our discussion is that the extremes in solar Beta Angle values represent special conditions for day versus night overflights at the extreme latitudes of the orbit.
In other words the sunny side of the orbit can be either at the high latitudes or the low latitudes, but not both in the same day. It takes half of the 46 day cycle, 23 days, to transition between these two extremes.
7. PASS TIMES SEPARATION VERSUS LATITUDE
Passes at latitudes below 35 North will lead or follow the local times for 35 north by a certain amount for ascending and descending orbit segment overpasses, respectively. For example, on a day when Charlotte North Carolina is getting Noon overflights, a site in the Florida keys may get a 9:00 a.m. overflight from the ascending portion of an orbit, and a 3:00 p.m. overflight four orbits later from the descending portion of the orbit. The "average" time of passes in the northern hemisphere follows the same pattern as noted above for 35 North, but passes will be offset from this "average" time as a function of latitude.
Based on prelaunch model of the orbit path, we estimate the following offset from the mean time of day for overflights as a function of latitude:
Latitude |
Ascending Phase Offset |
Descending Phase Offset |
|---|---|---|
| 34.0 | -1 hour | +1 hour |
| 31.0 | -2 hours | +2 hours |
| 25.0 | -3 hours | +3 hours |
| 18.0 | -4 hours | +4 hours |
| 7.0 | -5 hours | +5 hours |
More details of this relationship can be calculated based on the mission orbit characteristics as needed.
8. MORE INFORMATION
Questions about the orbit modelling and the overflight predictions can be
directed to Steve Bilanow at
