Method
Show the reasoning
General physics explains the relationships. Cited civil instruments supply their own published detail. The boundary between them stays visible.
Personal educational project based on cited public sources. Not an official publication of my employer or of the agencies or companies discussed. Models are schematic; estimates and assumptions are identified.
Scope · Formulas · Assumptions · Checks
What the engine calculates
The orbit example gives a two-body period, altitude, geometric distance to nadir, and one-way vacuum light travel time. Separate radiation examples calculate the energy of a photon, ideal circular-aperture diffraction, and the radiance of an ideal blackbody over a stated wavelength interval.
The teaching inputs do not specify the spacecraft in the drawing. Orbit choice affects the orbit example. Wavelength affects the optical and radiation examples. Changing orbit or detector material does not change the blackbody source radiance. A laboratory aperture is used only for the independent diffraction example.
The civil aperture selection leaves diameter and diffraction unavailable because the ABI aperture was not verified in the opened sources. Detector material selections do not assign a temperature. Published TIRS-2 temperatures describe Landsat 9 TIRS-2 specifically.
The engine contains no real plume intensity, received sensor photon count, detection threshold, signal-to-noise ratio, ground resolution, array format, frame rate, or cryocooler power model. It does not calculate constellation coverage, revisit, operational status, or warning latency. Atmospheric absorption is explained qualitatively; no transmission curve is assumed.
Formulas and units
Calculations use meters, seconds, kelvins, and exact SI radiation constants internally. Display values are rounded for reading. Each formula below links to its published physics and the teaching assumptions used with it.
Calc.
Position on an illustrative ellipse
Advance mean anomaly M uniformly in the illustration clock, solve M = E − e sin E, then use x = a(cos E − e), y = a√(1 − e²)sin E. The focus is the origin and positive x points to pericenter. Angles are radians. Drawing parameters and playback time are not measured spacecraft states; they do not replace the separate orbit examples.
Inputs: authored illustrative semi-major axis and eccentricity; illustration clock and phase; NASA orbital motion and JPL generic coordinate formulas.
Calc.
Refrigerator energy balance
Qhot = Qcold + Win for a complete cycle with no net stored-energy change. Qcold and Win are nonnegative energy magnitudes into the cooler; Qhot is the heat rejected. All terms must use the same energy unit, or all may be steady average powers. This follows from the first law, ΔU = Q − W, using work done on the cooler as positive input. The equation alone does not determine cooling capacity, temperature, efficiency, or electrical power.
Inputs: heat removed and work supplied in matching units; NASA first law and sign conventions; no net stored energy over the cycle.
Calc.
Two-body orbital period
T = 2π√(a³/μ), where a is the ellipse’s semi-major axis (the center-to-center radius for a circular orbit). The satellite mass is neglected. GEO uses the public reference altitude; other choices use the stated teaching geometry. Earth is treated as spherical; perturbations are omitted.
Inputs: semi-major axis; Earth GM from NASA/JPL; model-orbits assumption.
General orbit examples
Calc.
Two-body orbital speed
v = √[μ(2/r − 1/a)], where r is distance from Earth’s center and a is the semi-major axis. The result is an Earth-centered inertial speed, not speed over the rotating ground. A circle has r = a and v = √(μ/r). HEO is evaluated at apogee, r = a(1 + e). The calculation uses the physical teaching orbit, never the compressed drawing coordinates. Gravity is the only modeled acceleration; perturbations and maneuvers are omitted.
Inputs: physical semi-major axis and orbital radius; Earth GM from NASA/JPL; NASA JSC vis-viva Eq. (1); model-orbits assumption.
Calc.
Altitude on an ideal ellipse
r = a(1 − e²)/(1 + e cos ν); altitude = r − R. Perigee and apogee use a(1 − e) − R and a(1 + e) − R. The HEO teaching example is evaluated at apogee, not averaged over time.
Inputs: semi-major axis and eccentricity; true anomaly; spherical Earth radius.
General orbit examples
Calc.
Geometric distance to nadir
d² = r² + R² − 2rR cos ψ. The displayed example sets ψ = 0, so d = r − R. This is distance to the point directly below the orbit position, not a ground station link, coverage calculation, or actual observation.
Inputs: orbital radius; Earth radius; assumed nadir geometry.
General orbit examples · Reference distance
Calc.
One-way light travel time
t = d/c, with c = 299,792,458 m/s. Only vacuum propagation is counted. It is not warning latency; processing, communications routing, and other delays are absent.
Inputs: geometric nadir distance; exact SI speed of light.
General orbit examples · Reference distance
Calc.
Ideal circular-aperture diffraction
θ ≈ 1.22 λ/D radians: the Airy first-minimum angular radius for an unobstructed circular aperture. D = 0.30 m is a stand-alone laboratory example. No real payload aperture, image resolution, pixel size, or ground spot is inferred.
Inputs: model-lab-aperture assumption; model-band-examples assumption.
Ideal laboratory aperture · Teaching wavelengths and integration bounds
Calc.
Energy per photon
E = hν = hc/λ. Wavelength is converted from micrometers to meters. The calculation uses the exact SI values of h and c.
Inputs: illustrative wavelength; NIST h and c.
Teaching wavelengths and integration bounds
Calc.
Ideal blackbody radiance in a teaching band
Bλ = 2hc²/[λ⁵(exp(hc/(λkT)) − 1)] in W m⁻² sr⁻¹ m⁻¹. Integrate Bλ dλ over the assumed top-hat wavelength interval. Composite Simpson quadrature uses 512 intervals in log wavelength, including the λ Jacobian. This is an ideal 288 K blackbody, not the observed Earth or a plume.
Inputs: NIST h, c and k; model-blackbody assumption; model-band-examples assumption.
Ideal thermal source · Teaching wavelengths and integration bounds
Calc.
Ideal blackbody photon radiance
Integrate Bλ/(hc/λ) dλ over the same teaching interval. Units are photons s⁻¹ m⁻² sr⁻¹, not photons at a detector. No aperture area, field of view, atmosphere, throughput, quantum efficiency, or detector response is applied.
Inputs: Planck spectral radiance; photon energy at each wavelength; assumed teaching band.
Ideal thermal source · Teaching wavelengths and integration bounds
Calc. · reference formula only
Ideal refrigerator coefficient of performance
COPCarnot = Tc/(Th − Tc), with absolute temperatures and Th > Tc > 0. This limit is not the efficiency or input power of a real cryocooler; the scenario does not assign such values.
Inputs: cold and hot temperatures supplied to the pure function; NIST refrigeration review, Eq. 1.
The pure mathematics module includes this limit. The scenario assigns no cold-stage load, actual efficiency, cooler input power, or instrument temperature.
Teaching and drawing assumptions
These choices make the equations readable. They remain labeled Assumed even when a calculation uses them. A numerical result does not make its chosen inputs into measured hardware properties.
Assumed
Schematic orbit playback
Compressed drawing geometry and an illustration clock
Orbit guides retain their authored drawing dimensions and shapes. Motion follows each guide’s own ellipse or circle. The physical teaching orbit calculations are separate: no real constellation state, observation geometry, or operational schedule is inferred from this animation.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
Cooler energy accounting boundary
Complete cycle or steady average; no net stored energy
The symbolic balance treats the cooler as a closed cyclic device with heat absorbed, input work, and rejected heat. It assigns no numerical load, efficiency, temperature, or rate to a spacecraft. Startup, cooldown, and other transients would require a stored-energy term.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
General orbit examples
A spherical Earth and negligible satellite mass
The sphere uses the published equatorial radius, 6,378.137 km. GEO uses a 35,786 km circular reference altitude. HEO is a teaching ellipse with period one half of the published sidereal day and eccentricity 0.722, evaluated at apogee. MEO and LEO are arbitrary circular examples at 20,000 km and 1,000 km. These are not program or spacecraft orbital parameters, and no constellation is computed.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
Reference distance
The surface point directly below the illustrated orbit position
The central angle is zero. This keeps the distance and light-time example simple without assuming a real observing direction or ground station. HEO uses the apogee position.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
Ideal laboratory aperture
0.30 m unobstructed circular diameter
An arbitrary mathematical example used only for diffraction. It is not the aperture of any depicted military hardware or civil instrument. The civil aperture selection returns unavailable because the ABI diameter has not been verified.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
Teaching wavelengths and integration bounds
SWIR: 2.0 μm, interval 1.5–2.5 μm; MWIR: 4.3 μm, interval 3.8–4.8 μm; LWIR: 10.0 μm, interval 8–12 μm
These illustrative wavelengths and top-hat integration bounds are chosen to compare the equations. They are not formal definitions of the broad infrared band names and are not any instrument’s spectral response or atmospheric transmission windows.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
Ideal thermal source
288 K, emissivity 1
A uniform ideal blackbody provides an elementary radiance example. It is not a measured Earth scene, plume, satellite background, or detector background. Orbit, aperture and detector choices do not affect this source radiance.
The following references supply the underlying physics or published reference inputs. The choice of teaching inputs remains an assumption.
Assumed
Illustration geometry
Representative / not to scale
The prototype’s geometry, layout, satellite counts and positions, and ray paths are chosen to explain component roles. They do not describe a real constellation, sensor performance, or physical dimensions.
The scene colors and exploded spacing separate component roles. Amber light paths and blue cold parts are visual cues, not measured radiance or temperature. Geometric scales may be compressed or enlarged for legibility.
How the work is checked
The numerical tests check the orbital reference values, ellipse geometry, and vacuum light time. Radiation tests check wavelength units, the long-wavelength limit, integrated blackbody energy, photon-energy conversion, and convergence when integration intervals are refined. Tests also check every declared scenario combination and preserve unavailable civil and detector values.
The claim audit follows the same registry used by the cards and source dialogs. It rejects missing evidence, unknown calculations or assumptions, legacy labels, and unchecked citations. Type checks and browser gates cover interface behavior, framing, camera paths, rendering, and links.
Read the claim and source registers →