Perturbations, TLEs & Time

This section walks through Homework 2 the way you should approach it in an exam: identify what is given, choose the relation that connects the given to the unknown, and check the result against something you already know. The aim is that you can reproduce the reasoning, not that you can recall the final number.

Given / find: Find the classical orbital elements of a geostationary satellite, its apogee and perigee radii and heights, and its radial (angular) and linear speeds.

Approach: The three geostationary conditions from Week 3 — circular, equatorial, prograde — fix most of the elements immediately. Only the size has to be computed, and it follows from the period through Kepler's third law. Nothing here needs a look-angle formula.

Given / find: For Riyadh, Cairo, Casablanca and New York, find the slant range, elevation and azimuth to satellites at \(20^\circ\text{E}\), \(26^\circ\text{E}\) and \(30.5^\circ\text{E}\).

Approach: Test visibility first. Computing look angles for a satellite that is below the horizon wastes time and produces a meaningless answer, so the visibility check is step one, not a check at the end.

Each visible cell reads slant range \(d\ (\times1000\ \text{km})\) · elevation · azimuth.

Given / find: One earth station at latitude \(30^\circ\text{S}\) and another at \(30^\circ\text{N}\) share the same longitude, and the satellite is \(20^\circ\) east of both. Find the look angles for each station and the round-trip time, counting propagation delay only.

Approach: Use the symmetry before reaching for the equations. The two stations are mirror images about the equatorial plane, so the geocentric angle \(b\), the slant range \(d\) and the elevation are identical for both. Only the azimuth differs, because one station looks north of east and the other looks south of east.

Given / find: Find the fixed angle of tilt for a polar-mount antenna at your home location, taken here as latitude \(\lambda_E=25^\circ\).

Approach: A polar mount rotates about one axis only, so its elevation is set once and then left alone. That fixed elevation \(El_0\) is the polar-mount relation from Week 3, not the general look-angle elevation formula.

These are not specific to HW2. They are the recurring errors from Weeks 1 to 3, and they are worth checking before you submit anything.

Everything in Weeks 1 to 3 rests on one idealisation: two point masses, one gravitational force, nothing else. Under that assumption the orbit is a fixed ellipse in inertial space, and the six classical elements are constants. Only the true anomaly \(\nu\) changes with time.

Real satellites do not behave that way for long. Earth is not a point mass and not a sphere, the atmosphere does not stop abruptly, and the Sun and Moon pull on the satellite as well. Each of these adds a small acceleration on top of the two-body term:

The perturbing acceleration is tiny compared with the main gravitational term — often smaller by a factor of \(10^{3}\) to \(10^{6}\). That is exactly why the two-body model is such a good first answer, and also why it is misleading: a small acceleration integrated over many orbits becomes a large position error.

This is why a prediction that is excellent for one orbit can be useless after a few weeks, and why tracking data must be refreshed. It is not that the mathematics of Weeks 1 to 3 is wrong; it is that the model those equations describe is incomplete.

An orbit perturbation is any acceleration acting on the satellite that the two-body model does not include. The word is deliberate: these are not corrections to a mistake, they are physical effects that the idealised problem left out.

The useful consequence is that the six classical elements stop being constants. Instead of describing a fixed ellipse, they become slowly varying quantities — the orbit is still an ellipse at any instant, but that ellipse drifts, rotates and changes size over time. Engineers often describe perturbations by how the elements respond:

For this course the important distinction is the first one: secular effects are the reason an ideal propagation degrades with time, and the reason a TLE has an epoch attached to it.

Four effects account for nearly all of the departure from two-body motion for Earth satellites. Which one dominates depends almost entirely on altitude.

A two-line element set is the standard published description of a satellite's orbit: two lines of exactly sixty-nine characters in a fixed-column format. It carries a set of mean orbital elements together with the time they describe and a drag term.

A full field-by-field decoder, including the fixed-column layout and the parsing traps, is available in the TLE explainer lab.

Mean motion gives the semi-major axis through the relation you already know:

The orbit and ground-track visualizer already in the course implements exactly the chain built up in Weeks 1 to 3. Nothing in it is hidden; it is worth knowing the stages because the same pipeline appears in every ground-track tool.

1. Orbital elements. Start from \(a,e,i,\Omega,\omega\) and the mean anomaly at epoch — the same information a TLE carries.
2. Two-body propagation. Advance the mean anomaly at the constant rate \(n\), then solve Kepler's equation \(M=E-e\sin E\) for the eccentric anomaly and convert to true anomaly. This is the step that assumes a perfect Keplerian orbit.
3. Position in the inertial frame. Place the satellite in the perifocal frame, then rotate into ECI with \(R_z(\Omega)R_x(i)R_z(\omega)\). In ECI the orbit does not move.

4. Earth rotation. Rotate from ECI to the Earth-fixed frame by the Greenwich angle \(\theta_g\), which advances at one revolution per sidereal day, \(86{,}164.0905\ \text{s}\).
5. Latitude and longitude. Convert the Earth-fixed position to a sub-satellite point, \(\phi=\arcsin(z/r)\) and \(\lambda=\operatorname{atan2}(y,x)\).
6. Ground track. Plot successive sub-satellite points on the map. The familiar westward drift of each successive pass is Earth turning underneath a fixed inertial orbit.

SGP4 is the propagator that TLEs were actually designed for. Substituting it would change one stage of the pipeline and leave the rest intact.

We are not implementing SGP4 in this course and the visualizer is not being changed. The point is to know which single stage carries the modelling assumption, and what upgrading it would buy.

Coordinated Universal Time is the global civil time reference. It is generated from atomic clocks, and it is kept within a second of Earth's actual rotation by inserting occasional leap seconds. Local time zones are defined as offsets from it.

Satellite work needs a common time reference for an obvious reason: a position is meaningless without the instant it applies to. Ground stations on different continents must agree on when a pass begins, ranging measurements must be timestamped consistently, and orbit determination combines observations taken hours apart from different sites. All of it is done in UTC.

This is also how a TLE anchors itself. The epoch field is a UTC instant, written as a two-digit year followed by the day of the year including the fraction of a day. An epoch of 24145.51782528 means day 145 of 2024, at 0.51782528 of a day after midnight UTC — a little after 12:25 UTC.

Calendars are hostile to arithmetic. Months have different lengths, leap years interrupt the pattern, and subtracting two calendar dates requires special-case logic. Orbital mechanics needs to do arithmetic on time constantly — how many seconds since epoch, what is the interval between two observations — so it uses a continuous count instead.

The Julian date is that count: a single number giving the days, including the fraction of a day, elapsed since a fixed reference instant. Because it is just a number, the interval between two times is a subtraction, with no calendar logic at all.

In practice you convert the TLE epoch to a Julian date once, and every later question — how many seconds have elapsed, what is Earth's rotation angle now — becomes ordinary arithmetic.

Weeks 1 to 3 put the orbit in an inertial frame: ECI does not rotate, so the orbital plane stays fixed relative to the stars. A ground track, however, is drawn on a rotating Earth. Converting between the two viewpoints requires knowing how far Earth has turned, and that angle is what sidereal time measures.

The distinction matters because Earth's rotation relative to the stars is not the same as its rotation relative to the Sun. In one day Earth also moves along its orbit around the Sun, so it must turn slightly more than one full rotation for the Sun to return to the same meridian. The solar day is therefore about four minutes longer:

An orbit is fixed with respect to the stars, so it is the sidereal rate that governs how the Earth slides beneath it. The specific quantity needed is Greenwich Mean Sidereal Time — the angle \(\theta_g\) from the vernal equinox, which is the ECI \(x\)-axis, round to the Greenwich meridian, which is the Earth-fixed \(x\)-axis. Knowing \(\theta_g\) is exactly knowing the rotation between the two frames:

The three time concepts are not separate topics. They are consecutive links in one chain that connects a clock reading to a ground track, and every satellite tracking tool walks it.

1. UTC
2. Julian date
3. sidereal time
4. Earth rotation angle
5. ECI → Earth-fixed
6. latitude & longitude
7. ground track

Steps one to three are time-keeping, step four is a coordinate rotation, and steps five and six are geometry. The orbital mechanics of Weeks 1 to 3 supplies the inertial position that enters at step four.