Tracking issues for "the Great Pumpking" Tracking issues for "the Great Pumpking"
NOTE: This text is HTMLized edition of my paper at the "Gigaleiri", the annual Finnish higher frequencies radio amateur meeting held this year (1999) at the "Artjärvi Telecommunication Society" amateur station at Artjärvi, Finland.

I am member of that club.


Tracking issues for "the Great Pumpking"[1]

Abstract:

To be able to point a large, high accuracy radio telescope to desired direction, many things need to be taken into consideration.

Here are some discussions on the issues needed for pointing in different working skenarios. We begin with tracking speeds related to AZ-EL tracking at different stellar object declinations at our latitude of 60 degrees north.

Rest of the presentations cover issues of when to choose which of the correction algorithms, and in the end, how to do EME tracking in this type of system.

This presentation is full of possibly strange sounding terminology taken out of astronomy, which, after all, is basis for all the hard data, and methodology for this.[2]

About the telescope:

The telescope in question has diameter of 13.4 meters, and it can be used up to about 100 GHz (47 GHz being likely maximum). For tracking the criteria is to be able to track within 1/10 of the beam width at 100 GHz.

This criteria gives 1.2 millidegrees for the required pointing.

This criteria is also 1/300 000 of the full circle, so to be able to get feedback which is in par, or better than the desired pointing, at least 19 bits per full circle is required.

If the telescope does not track, but is pointing at the meridian (to the south, for the northern hemisphere people) at the declination of 0 degrees, the mere rotation of the earth at rate of 4.17 millidegrees per second will sweep this 1.2 millidegrees thru in mere 0.287 seconds.

That is, to maintain the tracking, the system can't tolerate lapses of practically any kind.

Therefore the likely processing model is to have semi-intelligent servo controllers, to which main coordinate controller just tells the direction (left/right), and the speed (nn nanoradians per second), and then uses a feedback loop of ``at that time, being at that point, and moving to that direction with that speed''. Likely controller in question is a form of PID (Proportional Integrator Derivator), which manages the real-time direction and movement speed.

AZ-EL tracking speed calculations at latitude 60 degrees North:

First of all, at a AZ-EL system the speed in the axis varies a lot, and there is a definite infinity point of the AZ speed for objects passing thru the zenith.

Fastest tracking speeds in degrees per minute for given declination:

      decl     EL range          delta AZ          delta EL

     -30.0  -60.0 ..  0.0   0.1966 .. 0.4330  -0.1250 .. 0.1250
     -10.0  -40.0 .. 20.0   0.2143 .. 0.3214  -0.1250 .. 0.1250
       0.0  -30.0 .. 30.0   0.2165 .. 0.2887  -0.1250 .. 0.1250
      10.0  -20.0 .. 40.0   0.2143 .. 0.3214  -0.1250 .. 0.1250
      20.0  -10.0 .. 50.0   0.2077 .. 0.3655  -0.1250 .. 0.1250
      30.0    0.0 .. 60.0   0.1966 .. 0.4330  -0.1250 .. 0.1250
      40.0   10.0 .. 70.0   0.1808 .. 0.5599  -0.1250 .. 0.1250
      50.0   20.0 .. 80.0   0.1587 .. 0.9254  -0.1250 .. 0.1250
      55.0   25.0 .. 85.0   0.1434 .. 1.6449  -0.1250 .. 0.1250
      58.0   28.0 .. 88.0   0.1302 .. 3.7910  -0.1250 .. 0.1250
      59.0   29.0 .. 89.0   0.1237 .. 7.3391  -0.1250 .. 0.1250
      59.5   29.5 .. 89.5   0.1191 .. 14.246  -0.1250 .. 0.1250
      60.0   30.0 .. 90.0    -INF  .. 0.1443  -0.1250 .. 0.1250
      60.5   30.5 .. 89.5  -13.826 .. 0.1429  -0.1231 .. 0.1231
      61.0   31.0 .. 89.0  -6.9094 .. 0.1414  -0.1212 .. 0.1212
      62.0   32.0 .. 88.0  -3.3588 .. 0.1384  -0.1174 .. 0.1174
      65.0   35.0 .. 85.0  -1.2120 .. 0.1290  -0.1057 .. 0.1057
      70.0   40.0 .. 80.0  -0.4924 .. 0.1116  -0.0855 .. 0.0855
      80.0   50.0 .. 70.0  -0.1269 .. 0.0675  -0.0434 .. 0.0434
      85.0   55.0 .. 65.0  -0.0516 .. 0.0380  -0.0218 .. 0.0218
      90.0   60.0 .. 60.0   0.0000 .. 0.0000   0.0000 .. 0.0000

    decl:  object declination in sky, degrees
    EL:    elevation above horizon (lowest .. highest, degrees)
    dAZ:   maximum AZ speeds in degrees per minute (left .. right)
    dEL:   maximum EL speeds in degrees per minute (down .. up)

This clearly shows that we should not track within a few degrees of the zenith. If we put speed limit at 1 degree per minute for the high accuracy tracking, then the limit distance is something like 9 degrees from the zenith. If we allow up to 3 degrees per minute (which is about 10 times the declination 0 AZ speed at the meridian passage), we can go up to about 2.5 degrees from the zenith.

Doing the basic geocentric RA/DEC conversion to local AZ/EL coordinates takes around 20 microseconds at my Cyrix 6x86 running at about 100 MHz. That is, if all the system does is to do these conversions, then it can do them about 50 000 in a second.

A thing not visible here is, that the sign of the Elevation change changes at the meridian passage, and depending on the drive mechanism, that may require a moment, during which no accurate tracking is possible, due to slacks in the pedestal gear trains.

Needs for topocentric correction:

At least for EME uses we need to do correction for topocentric coordinates of given spot at the surface of the moon (e.g. 47 GHz beam width is mere 27 millidegrees, which is about 18th part of the visible moon dish.)

In theory the pointing criteria will force us to need topocentric coordinate correction for all objects nearer than 1/10th or about 1.2 millidegrees from earth/moon system gravitational middle point. Longest distance a point of earth's surface can be from that point is about 10 000 kilometers.

[picture explaining parallax]

Multiplying that 10 000 km by 48 million (which comes from parallax of the 1.2 millidegrees -- roughly sin(1.2 millidegrees), or 1/48 000 000) we get nearest distance where the effect of the topocentric correction is same or less than our criteria: about 477 000 000 000 km, or 3200 Astronomical Units. (For comparison: Earth to Sun is 1 AU, Pluto is "only" about 40 AUs away! One Light Year is about 63 000 AUs.)

At 10 GHz the beam is 10 times wider giving the longest distance for topocentric correction need of 320 AUs.

E.g. in practice all tracking within the solar system will need it. Stellar/Galactic tracking won't need the correction.

For a comparison, NASA's Tinbinbilla, Madrid and Goldstone trio operate down to 13 mm or 23 GHz, which with diameter of 64 meters gives beam width of 11.2 millidegrees, thus they need to do the full topocentric handling in pointing, too. -- that is what the used sources did tell about these radio-dishes capabilities, but likely the area for "23 GHz good" surface is smaller than 64 meters..

When to calculate which of the topocentric corrections:

For moon, we could use special model entirely counting telescope, and the desired beam middle point coordinates in 3D space (along with movement vectors). The Earths geodetic ellipsoid at latitude 60N is about 18.5 km off the ideal ball surface, and the geocentric latitude is roughly 0.166 degrees south from the ideal ball surface coordinates. This correction is visible in the results until about 6 Astronomical Units distance. (Per the 1.2 millidegree criteria.)

One way around could be to determine the true 3D orthogonal position X/Y/Z offset for the telescope at the surface of the earth in comparison to earths calculated centre point, and do parallax correction from that to the target point in the previously calculated earth-centric co-ordinates. This requires naturally also the distance to the target.

Local geometric corrections:

The telescope mechanics will, highly likely, have some sort of mis-alignment; the AZ-axis isn't exactly in local vertical, or the EL-axis isn't exactly horizontal. Also likely parts of the gear mechanics contain some non-idealities, which will appear as periodic and/or direction dependent errors.

This type of corrections are hopefully tabluable in AZ/EL matrix, which tells how much the error is in which direction at which part of the sky.

Errors difficult to fix relate to need to change tracking direction in the middle of the run, or caused by differential thermal expantion of the structures of the telescope during environmental temperature changes.

Aberration caused by relative speed of the telescope in relation to the direction of the object:

Essentially the angle (in radians) is about (v/c)*sin(theta), where ``theta'' is the angle in between the movement vector of the telescope, and object. In order of the aberration to exceed 1.2 millidegrees, the speed of the earth (and thus the telescope) must exceed about 6 km/sec, which isn't all that much! Earth's orbital speed around the sun is about 30 km/sec. However the largest this aberration can be is thus about 6 millidegrees, which is half beam width at 100 GHz.

[picture of relativistic aberration -- sideways moving telescope]

This earth's orbital form of aberration is yearly, thus it is quite enough to calculate it at the beginning of the tracking session as a part of the stellar coordinate epoch transformations; presuming we don't run for months on end for the same far away target without any restarts/target changes.

At EME work, the relative speeds are in order of 1-3 km/sec, and thus this aberration does not matter. Earths rotation speed at the equator is about 0.46 km/sec, making less than 0.1 millidegrees worth of relativistic aberration.

Refraction:

No mentions in used sources told of the atmospheric refraction affecting the radiowaves. Anyway, optical (450 THz) vs. 100 GHz wave length is different by factor of 450, but the phenomena of refraction being dependent by photon's energy (wave length), the associated refraction could be by that much smaller. At the largest the optical refraction at the horizon seems to be roughly 500 millidegrees, giving reason to assume that at 100 GHz it is within the 1.2 millidegree criteria.

[picture of gradual refraction]

To be accurate, the atmospheric refraction is highly dependent on the temperature/humidity/pressure at different locations along the incoming path. However, for a term which can be ignored, an averaged table can also be used.

The end result: refraction can be ignored, or at most entered into local static geometry correction tables.

Effects of Earth's magnetosphere:

Magnetic fields don't affect the direction the photons are going to, but they may cause polarity rotation, and spectral line (frequency) changes (so called Zeeman effect).

EME and the moon:

For EME uses we can likely use "simple" Earth+Moon+Sun perturbation model where e.g. Jupiter is left out, which calculates central points of earth and moon, plus libration model to calculate the orientation of the moon at its orbit. Then we map the telescope location at earth ellipsoid, and the desired beam middle point at the moon surface coordinates. (12 millidegree beam width at farthest away position along the path of the moon at the moon's apogee point is about 85 km, or about 1/40th of the moon diameter.)

A slightly simplified model does not care about moon's libration, and uses idealized spherical moon, which has its 0/0 coordinate point towards the earth/moon gravitational middle point, thus the task of agreeing at two EME stations to track at a given point at moon surface would be slighly simpler..

[a picture describing effects of the parallax in EME.]

In all cases, we have to agree with the opposite station on where at the moon surface the desired beam middle-point shall be placed, or the stations will miss each other's beams (assuming alike quality stations.)


References:

  • [1] "The Great Pumpking" reference in Artjärvi Telecommunication Society pages
  • [2] Fundamentals of Astronomy by Karttunen, Donner, Kröger, Oja, Poutanen; 2nd Finnish Edition: "Tähtitieteen Perusteet, URSA"
  • [3]A bit of math required for the beam-width calculation: beam-width in radians in case the wavelength is sufficiently much smaller than the diffraction aperture (radio dish diameter) is: Lambda/Diameter.