Pass geometry and Doppler for small-satellite ground stations

The orbit in numbers#

We take the orbit that many PocketQubes and CubeSats share: circular, 525 km, Sun-synchronous, the orbit of the Transporter rideshares that carried Grizu-263A and Delfi-PQ[1].

The satellite circles the Earth 15 times a day at 7.6 km/s. A ground station sees it only when it rises above the local horizon, or above the elevation mask the station sets to clear buildings, trees and the noisiest low-angle directions.

How far, and for how long#

Seen from the centre of the Earth, the station and the point under the satellite are separated by the central angle $\lambda$. The satellite is visible above elevation $\epsilon$ while[2][3]

At 525 km that is 22.5° for the horizon, a circle of about 2500 km radius on the ground, and 1600 km for a 10° mask. The distance to the satellite, the slant range, follows from the triangle formed by the Earth’s centre, the station and the satellite:

The longest pass, with the satellite going straight overhead, lasts the fraction $2\lambda /2\pi$ of the orbital period: 11.9 minutes above the horizon and 7.7 minutes above 10°. Most passes are shorter, because the ground track misses the station.

Elevation during a pass#

Elevation rises slowly at the start, sweeps through its maximum quickly and falls slowly again. An overhead pass spends only 3.6 minutes above 30°. For a station with an azimuth–elevation rotator, a pass near zenith is the hardest case: the azimuth has to swing through almost 180° within seconds around culmination, faster than many rotators can follow. Prediction software usually flips the antenna, or the station accepts a short loss of pointing.

The Doppler shift#

A signal from a source moving along the line of sight at range rate $\dot{d}$ is received shifted by

The shift is largest at the horizon, where the line-of-sight component of the orbital velocity $v$ is $v{R}_E/(R_E+h)$, and zero at closest approach. At 525 km and 437.19 MHz the satellite arrives about 10.2 kHz high and leaves 10.2 kHz low, a total sweep of 20.5 kHz.

The curve is steepest at the moment of closest approach. For an overhead pass the range acceleration there is $\ddot{d}=v^2{R}_E/[h(R_E+h)]$, and the largest Doppler rate is11From $d^2=a^2+R_E^2-2a{R}_E\cos \omega t$ with $a=R_E+h$: differentiating twice and setting $\dot{d}=0$ at $t=0$ gives $\ddot{d}=a{R}_E{\omega }^2/h$, and $a\omega =v$.

which is 148 Hz/s at 437 MHz and 525 km. A lower pass is gentler, because the closest distance is larger. Both the shift and the rate scale with frequency:

Why a narrowband link cares#

A ±10 kHz drift moves a narrow telemetry signal by about its own bandwidth or more. A receiver has two choices. It can open its filter wide enough to keep the signal inside for the whole pass, at a price: every doubling of the noise bandwidth costs 3 dB of signal-to-noise ratio, more than the margin of a typical PocketQube link budget. Or it can follow the predicted Doppler curve, which ground-station software does by retuning continuously from the orbital elements. Open networks such as SatNOGS work this way, recording each scheduled pass with Doppler correction[4], and decoders such as gr-satellites then demodulate the recordings[5].

The uplink has the mirror problem. A small satellite’s receiver rarely searches in frequency, so the ground station pre-corrects its transmit frequency by the Doppler shift the satellite will see.

Coherent demodulators add a second requirement: their carrier loop must follow the Doppler rate, 148 Hz/s at UHF, 760 Hz/s at S-band and close to 3 kHz/s at X-band on an overhead pass.

Doppler as a measurement#

The Doppler curve carries information about the orbit. Its zero crossing marks the time of closest approach, and its slope there gives the closest distance through the rate formula above. A single well-recorded pass therefore measures where the satellite is along its track and how close the ground track came to the station.

That matters most right after a rideshare, when dozens of objects are released within minutes and catalogue elements cannot yet be assigned to individual satellites. Fitting the Doppler curves of individual passes resolves which object is which[1]: Grizu-263A was tracked under a temporary identifier for days before it received its NORAD number. It matters again late in a mission, because elements degrade for light spacecraft as drag grows[6]. Timing errors appear directly as frequency errors near closest approach: elements that place the satellite 10 s late, 76 km along its track, misplace the received frequency by up to 1.5 kHz at 437 MHz. In the waterfall images that SatNOGS archives for every observation[7], this shows up as a trace that drifts off the predicted curve.

flowchart LR
  A["Orbital elements"] --> B["Pass prediction"]
  B --> C["Rotator and Doppler schedule"]
  C --> D["Recording of the pass"]
  D --> E["Demodulated frames"]
  D --> F["Measured Doppler curve"]
  F --> G["Time of closest approach, closest range"]
  G --> H["Refined orbit, object identification"]
  H --> A

How much contact a station gets#

A Sun-synchronous orbit crosses each latitude at roughly the same two local times every day, once going north and once going south, so passes come in two clusters about twelve hours apart. How many fall in each cluster depends on latitude:

A mid-latitude station like the one in Zonguldak that operated Grizu-263A gets about three passes a day above 10°, some 20 minutes of contact, and waits up to 13 hours between clusters. At Svalbard, near the turning latitude of polar orbits, the same satellite passes 11 times a day. Lowering the mask to 0° at 41.45° N would raise the count to 5.5 passes and 50 minutes, but the added minutes are the low-elevation ones, where the link budget is weakest. Networks change the picture more than any single site: a few dozen stations spread over latitudes and longitudes turn minutes per day into hours.

Rules for tracking a small satellite#

1. Start from the best elements available. After deployment, use the deployer’s state vector or the first catalogue elements, and refine with Doppler fits until the object is identified.
2. Correct Doppler in software, on the downlink and the uplink, rather than widening filters.
3. Choose the elevation mask from the link budget, not only from the horizon: 10° is a common compromise at UHF.
4. Budget contact minutes, not passes. Three passes a day at mid-latitude amount to about 20 minutes above 10°, which bounds the data a single station can collect.
5. Plan for the overhead pass, which brings the strongest signal and the most demanding rotator motion.

The ground segment entry describes station hardware and networks, and the G/T article quantifies how much a better station buys.