Orbit Determination from Observations
Rendezvous includes a special panel which can load observational ephemeris data and estimate the orbital elements. These elements are listed in the Orbit Determination panel and can optionally be loaded into the main window for display on the map. There are several sample ephemeris files to demonstrate how to use the Orbit Determination panel to calculate orbital parameters. These ephemeris files contain measurements of the right ascention and declination of the target, the date and time of the measurement, and the latitude and longitude of the observer. GeoCentric Orbit Estimation We'll begin by calculating the orbit of IRAS (Infra-Red Astronomical Satellite) using measurements taken on January 27, 1999 from Colorado Springs. Be aware that these are not actual measurements. This ephemeris file contains simulated measurements of right ascention and declination calculated using the sat329.tle file. Follow these steps to load the ephemeris file and calculate the orbit: |
o | Click the Orbit Determination : Load Ephemeris button in the main menu | |
o | Navigate the Open Panel to the Ephemeris/IRAS_jan27.ephem file and click the Open button |
A panel appears which looks like this:
Each row in the scrollview contains a measurement of right ascension and declination. By default, all the data points will be used in the calculation of the orbit. You can selectively exclude data points by selecting them and selecting the Exclude selected Data button in the Data Exclusion Rule are of the panel. We will use all the data. The panel provides us a way to choose whether we wish to calculate the orbit as heliocentric (orbiting the sun) or geocentric (orbiting the earth). IRAS is a geo-centric satellite. |
o | Select the Earth Centered button in the Orbit Focus section of the panel | |
o | Click the Calculate State button at the bottom of the panel |
The orbit is now calculated and the orbital elements displayed in the State Vector section of the panel. The panel should now look like this:
We can display this orbit in the main window by pressing the Add To Main button at the bottom of the panel. When you do this, the orbital elements are entered into the main window as an entry in the Targets browser with the same name that is shown in the Name text field in the panel above. Double-clicking this new entry in the Targets browser will display the orbit in the Ground Trace Map. |
o | Click the Add To Main button in the panel | |
o | Double-click the entry IRAS_jan27 in the Targets browser of the main window. |
Let's check how well the calculation of the orbit matches the known orbit. We'll load the sat329.tle file into the Targets browser: |
o | Click the Constellation : Open Targets button in the main menu | |
o | Navigate the Open panel to the STATES/sat329.tle file, and click the OK button | |
o | Click the new sat329 entry that appears in the first column of the Targets browser of the main window | |
o | Scroll the second column of the Targets browser to the IRAS entry, and double-click on it | |
o | Set the Date to January 27, 1999 in the Calendar of the main window | |
o | Set the Clock to 12 : 25 : 13 in the main window |
The main window should look like this:
Even though we have two separate satellites selected, we can only see one. The orbital calculation is so close to the true orbit, we can't see the difference in the map. However, we can begin to see the orbits diverge if we advance the calendar. Let's look at the orbits after two days. |
o | Select day 29 in the calendar |
On the 29th of January, the orbits now look like this:
There is an apparent separation of the orbits now.
Error in Orbital Parameter Estimation Not all ephemeris data will yield such a good estimate of the orbit. Let's look at this example, where we use the last three data points in an ephemeris file: |
o | Load the ephemeris file MIR_jan12.ephem using the procedure outlined above | |
o | Select the last three data points in the scrollview of the Orbit Determination panel | |
o | Select the Include selected Data button in the Data Exclusion area of the Orbit Determination Panel | |
o | Select the Earth Centered button in the Orbit Focus area of the Orbit Determaination Panel | |
o | Click the Calculate State button at the bottom of the panel |
The Orbit Determination Panel should now look like this:
Let's add this orbit to the main window Target browser: |
o | Click the Add To Main button at the bottom of the Orbit Determination panel |
Set the date, time, and the length of the orbit in the main window: |
o | Set the Calendar to January 12, 1999 | |
o | Set the Clock to 13 : 25 : 35 | |
o | Enter the value 5500 in the End text field of the Ground Trace Control area of the main window |
Now, let's compare this orbit to the true orbit an Orthographic Projection. |
o | Deselect the IRAS_jan27 entry in the Targets browser of the main window (if it is selected) | |
o | Deselect the sat329 : IRAS entry in the Targets browser of the main window (if it is selected) | |
o | Select the MIR_jan12 entry in the Targets browser to display the calculated orbit | |
o | Select the sat329 : MIR entry in the Targets browser to display the true orbit |
Bring up the Map Inspector: |
o | Click the Inspectors : Map button in the main window | |
o | Click the Orthographic (Rendered) button in the Map Projection area of the Map Inspector panel | |
o | Enter the value 40 in the View Latitude text field of the Map Boundary area of the Map Inspector panel, and then click the tab button to move to the View Longitude field | |
o | Enter the value 10 in the View Longitude text field, and then click the carriage return key on the keyboard |
The map in the main window now looks like this:
While the two orbits start out very nearly in the same place, we can see the orbits diverge very quickly because the eccentricity which was calculated for this orbit (e=0.12) is substantially different than the true eccentricity (e=0.001).
HelioCentric Orbit Estimation We can also estimate the orbits of objects orbiting the sun such as comets and asteroids. In this section we'll use ephemeris data to find the orbit of Comet Hale-Bopp, and use the Solar System Inspector to view its path. The procedure is nearly identical to that for geo-centric orbits. |
o | Click the Orbit Determination : Load Ephemeris button in the main menu | |
o | Navigate the Open Panel to the Ephemeris/HaleBopp_jun96.ephem file and click the Open button | |
o | Select the Sun Centered button in the Orbit Focus section of the panel | |
o | Click the Calculate State button at the bottom of the panel | |
o | Click the Add To Main button at the bottom of the Orbit Determination panel |
Set the date and time in the main window: |
o | Set the Calendar to December 25, 1995 | |
o | Set the Clock to 18 : 00 : 00 |
Now lets add the rest of the planets to the Target browser: |
o | Click the Constellation : Open Targets button in the main menu | |
o | Navigate to the PolynomialPlanets entry | |
o | Click the Pull-down button in the Platform area of the Open panel, and select the PolynomialPlanet entry | |
o | Click the OK button at the bottom of the panel |
Select the Hale-Bopp comet and all the newly-loaded planets in the Targets browser: |
o | Double-click the HaleBopp_jun96 entry in the first column of the Targets browser in the main window | |
o | Double-click the PolynomialPlanets entry in the first column of the Targets browser in the main window |
Since these are all helio-centric orbits, they don't show up well in the Ground Trace map in the main window. For this reason, there is the SolarSystem Inspector which was created explicitly to show the orbits of planets, comets, and asteroids around the sun. Let's bring up a SolarSystem Inspector: |
o | Click the Inspectors : Solar System button in the main menu |
A panel appears which looks like this:
The default display will show objects inside a radius of about two astronomical units (that's the value in the Field of View text field). The (nearly) circular orbits are those of Mercury, Venus, Earth, and Mars. Comet Hale-Bopp has an orbit nearly perpendicular to the ecliptic plane (we can see that the inclination has a value of 89.04 in the State Vector area of the panel). The direction of the vernal equinox (the X direction in ECI coordinates) is shown by the bright red line projecting to the right. The Y axis is a slightly darker line pointing toward the top of the view. Click-dragging the mouse will change the point of view (note that the view won't change until after you release the mouse button). Zoom the map out: |
o | Enter the value 3.5 in the Field of View text field, and press the Carriage return key on the keyboard |
We can see a nice view of Hale-Bopp as it moves upward through the orbital plane of Jupiter by click-dragging the mouse:
Algorithm Used for State Vector Determination Rendezvous uses two algorithms for orbital determination calculations. A preliminary orbit is calculated using the method described in Fundamentals of Astrodynamics, by Bate, Mueller, and White in section 5.8. Further refinement is performed using the Differential Corrections algorithm described in the same book in section 2.12.2. While the preliminary orbit calculations do provide a satisfactory rough determination of a trial orbit, I am unsatisfied that the refinement procedure performs in an adequate manner. I am currently working on a Kalman filter to replace the differential corrections calculations. I hope this will allow more data to yield better orbits. |