Vol. Science · Astronomy · The coast
The cheap way from one circular orbit to another, in the classroom, is not a straight line. It is half an ellipse, tangent to the inner circle and tangent to the outer circle, with the central body at one focus. Two impulses do the work. The first burn leaves the departure circle. The craft coasts. The second burn joins the arrival circle. For circles of radii r₁ and r₂, the transfer semi-major axis is at = (r₁ + r₂) / 2. The coast lasts half an ellipse period, t = π √(at³ / μ). Speeds come from a circular orbit, v = √(μ / r), and from vis-viva, v = √(μ (2/r − 1/a)). Around the Sun, μ is the Sun Fact Sheet GM, 132,712 × 106 km3/s2. The radii are each planet’s orbital-parameters semimajor axis in 106 km, and the periods used for the phase angle are the sidereal orbit periods on those same NSSDCA fact sheets. Earth 149.598 × 106 km to Mars 227.956 × 106 km takes 258.88 days and 5.594 km/s. That is about 8.5 months of Earth’s 365.256-day sidereal year. The year of one circle is the Kepler desk. The speed to stay on a circle, or to leave a world, is the Escape desk. Speed at any r on one ellipse is the vis-viva desk.
Classroom schematic assuming circular, coplanar orbits and impulsive burns. It ignores planetary gravity wells and patched conics, inclination, eccentricity, and real mission design. It is not mission-planning software. Top-down, from the north, so motion runs counterclockwise. The Sun (or Earth, in the around-Earth case) sits at the focus of the transfer ellipse, not at the ellipse’s center. Planets are marks, not worlds drawn to size. Play is a time-lapse of the coast, about 8 seconds, not a clock. The 400 km low-orbit altitude in the around-Earth case is a round classroom example. The Earth Fact Sheet does not print it. The geosynchronous radius is computed from that sheet’s GM and its 23.9345-hour sidereal rotation period.
From the north · Sun at the focus · scrub is time
Departure
Arrival
Low orbit · example altitude, not a fact-sheet line
The formulas
at = (r₁ + r₂) / 2. The transfer ellipse is tangent to the inner circle at one end of its long axis and tangent to the outer circle at the other. Those two radii are the periapsis and the apoapsis, so the semi-major axis is their average. The central body is at the focus, a distance |r₂ − r₁| / 2 from the ellipse’s center.
t = π √(at³ / μ). A full ellipse has period 2π √(a³ / μ). The Hohmann coast is half of that, from one apsis to the other. μ is GM of the body at the focus. For planets, μ is the Sun Fact Sheet’s 132,712 × 106 km3/s2. For the around-Earth case, μ is the Earth Fact Sheet’s 0.39860 × 106 km3/s2.
v = √(μ / r) on a circle. v = √(μ (2/r − 1/a)) on the ellipse, which is vis-viva. At the inner end the ellipse is faster than the inner circle, so an outward trip speeds up. At the outer end the ellipse is slower than the outer circle, so the second burn speeds up again to join that circle. An inward trip slows down at both ends. Δv₁ and Δv₂ are the absolute differences. Total Δv is their sum.
θ = 180° − (360° × t / Ptarget). During the coast the target moves 360° × t / P. The craft moves 180° of the transfer. The target must already lead by θ at departure so the two meet at the far apsis. A negative θ means the target is behind the departure body, along the direction of motion. Ptarget is the fact-sheet sidereal period for a planet. Around Earth, P is the circular period 2π √(r³ / μ) from the same GM and radius. The window spacing is the synodic period |P₁ P₂ / (P₁ − P₂)|, from those same periods. When the formula’s θ passes ±180°, the drawing places the target at that angle modulo 360°. The printed θ stays the formula.
Literacy · a coast, not a mission
Sources · public NASA pages
Classroom picture of a Hohmann transfer. Earth to Mars, from the kilometer semimajor axes and the Sun’s GM above, is 258.88 days and 5.594 km/s. Circular, coplanar, impulsive. Not patched conics, not a launch, and not mission-planning software. The 400 km altitude is an example, not a fact-sheet line. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The year
P² = a³ in years and AU. How long one solar circle takes, not the coast between two.
The speed
Circular orbit and escape at one altitude. √2 apart. A cannon diagram, not this transfer.
The ellipse
The same formula, anywhere on one ellipse. Earth’s perihelion rounds to 30.29 km/s. Not an ephemeris.
Neighborhood
Where the eight planets sit. Day and local time, still a schematic.
The pull
The force that makes both circles, and the ellipse between them, fall.
Two circular orbits and the half-ellipse that touches both. The focus is the Sun, or Earth in the around-Earth case. The picture assumes one plane and instantaneous burns. It is not a trajectory to fly.
| Earth to Mars | The default. Fact-sheet semimajor axes 149.598 and 227.956 × 106 km. Transfer time 258.88 days. Total Δv 5.594 km/s. |
|---|---|
| Earth to Venus | An inward coast. Both burns slow the craft down. The phase lead comes out negative: Venus starts behind Earth. |
| Earth to Jupiter | A long outward coast. The inner circle is drawn to scale, so Earth is smaller on the disk than it is for Mars. |
| Departure and arrival | Any two of Mercury through Neptune. The radii are the fact-sheet kilometer semimajor axes. The same planet on both ends is not a transfer. |
| Around Earth | Leaves the Sun. μ is Earth’s GM, 0.39860 × 106 km3/s2. The outer circle of an outward trip is the geosynchronous radius computed from the 23.9345-hour sidereal day. The inner circle is the volumetric mean radius, 6,371.000 km, plus the altitude slider. |
| Altitude | 0 to 2,000 km above the volumetric mean radius. It starts at 400 km, a classroom example. The fact sheet does not print that altitude. Moving it changes the low orbit only. |
| Low to geosynchronous | Outward. Speed up at the low orbit, coast, speed up again to join the geosynchronous circle. |
| Geosynchronous to low | The return. Same ellipse, other way. Both burns slow the craft down. The Δv magnitudes match the outward trip. |
| Along the coast | Time along the transfer, from the first burn to the second. The craft moves faster near the inner orbit, because vis-viva says so. The slider is time, not an even angle. |
|---|---|
| Play | Runs the coast in about 8 seconds, then stops at arrival. Press again to start over. It is a time-lapse, not those days on a clock. Pause keeps the place. |
| Focus | The Sun, or Earth. The circles are centered on it. The ellipse is not. Its center sits toward the outer apsis. |
|---|---|
| Inner and outer circles | The two circular orbits, to scale with each other. A far arrival makes the departure orbit a small ring. That scale is the point. |
| Transfer | The bright half of the ellipse, the half the craft flies. The faint other half is the rest of the same ellipse, not a second route. An outward trip uses the upper half. An inward trip uses the lower half, so the coast stays counterclockwise. |
| Craft | The moving mark. At the start it sits on the departure body. At the end it sits on the target. |
| Phase arc | The angle at the focus between the departure body and the target at the first burn, drawn modulo 360°. The readout’s θ is the formula, which can pass a full turn when the target is much faster. |
| N | North, the view. Motion runs counterclockwise, the same sense as the other top-down desks. |
| Transfer time | π √(at³ / μ), in days. Under two days, the face switches to hours. The subtitle in days uses Earth’s 365.256-day sidereal period divided by 12, labeled as that quotient, not as a calendar month. |
|---|---|
| Δv₁ and Δv₂ | The speed change at departure and at arrival. “Speed up” is prograde. “Slow down” is retrograde. Total Δv adds the two magnitudes. |
| Circular and vis-viva | √(μ / r) on each circle, and √(μ (2/r − 1/at)) on the ellipse at that same r. |
| Phase lead | θ = 180° − (360° × t / Ptarget). Planet periods are the fact-sheet sidereal days. Around Earth, the periods are 2π √(r³ / μ). A second line appears when θ and the on-circle angle differ by a full turn. |
| Window spacing | |P₁ P₂ / (P₁ − P₂)| from those same periods. How often this phase repeats. Not a launch calendar. |