Solar System Orbits — Ellipses & Speeds

Source

Lecture: L1 — Classical mechanics as prediction via differential equations
Simulation: 2-D top-down solar-system orbits (Guided Inquiry)
Concept: Planetary orbits are ellipses with the Sun at a focus, traversed at varying speed.
Serves: Framing: predicting the future by solving Newton's equations numerically (velocity Verlet).
Inquiry: Where on its orbit does Mercury move fastest? Which orbit is least circular?
Misconception: That planetary orbits are circles traversed at constant speed.
Units: Distances in AU, time in years, so GM_☉ = 4π² AU³/yr². Speed converted to km/s for display (1 AU/yr ≈ 4.74 km/s). Full-system view: spacing between orbits is compressed for legibility; each orbit's shape, eccentricity, and Sun-at-focus placement are exact.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Structural misconception targeting	Wrong "circular / constant speed" idea is encoded as a prediction choice; if chosen, a dashed ghost planet moves at constant angular speed alongside Mercury so the misconception visibly fails.
H2	No standing misconception prose	The rebuttal is revealed only when the student selects the wrong choice.
A1	One concept, one sentence	Orbits are ellipses; speed varies (faster near perihelion).
A2/C1	Numbers earn their place	r/v/a/e appear only when the step that reads them activates.
E1	Physics honesty	Velocity Verlet (symplectic) integrates real GM/a/e; no Euler drift; scale convention printed on canvas.
D6/D7	Scene evolves with steps	Planet list and stats (including e) reveal as the student advances.
A3/A4	Formal tools off by default	vis-viva and Verlet equations live behind ⚛︎ The Physics (the pop-up revision sheet); the sim reads without them.

Solar System Orbits

Teach me

Time rate

Orbital view — 3D — Inner system (out to Mars)

drag to pan, scroll to zoom

Guided Inquiry

1 · The scene

Let’s get our bearings. The yellow disk at the centre is the Sun. Each coloured dot is a planet, and the closed curve through it is that planet’s orbit; Mercury is the innermost.

The distances you see are to scale, but the planet icons are enlarged so they stay visible.

2 · Speed around the orbit

Let’s follow just one planet. Mercury runs once around its orbit. Along that path, where do you think its speed is greatest?

3 · Distance and speed together

Now let’s watch it happen. Mercury is selected: r is its distance from the Sun and v is its speed.

Follow the pair through one full lap. When r shrinks, what does v do — and what happens out on the far side? Keep your eye on both numbers.

4 · Where the Sun sits

Here’s a subtler one. The Sun lies inside every orbit drawn here. In which one do you think it sits noticeably away from the centre?

5 · Far out

Now let’s head way out. Neptune’s orbit is about 30 times wider than Earth’s. What would you guess — compared with Earth’s orbital speed, Neptune’s is:

6 · Across the whole family

Now it’s over to you. All eight planets are in view, so select a few and compare each orbit’s size a with that planet’s speed v and its shape e.

As a grows, what happens to v? See if you can spot the pattern.

Listen

View

▾

Zoom

Highlight planet

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Selected planet

▾

Name—

a (semi-major axis)—

e (eccentricity)—

r (now)— AU

v (now)— km/s

Newton's law of gravitation

Units: r in AU, t in yr → GM_☉ = 4π² AU³/yr².

vis-viva — speed vs distance

v in AU/yr; multiply by 4.74 to get km/s.

Velocity Verlet (symplectic integrator)

h = integrator timestep (yr); sub-stepped so Mercury stays stable at fast time-rates.

Ellipse geometry

θ measured from perihelion; Sun sits at one focus.

The Physics

Solar-system orbits · what this sim shows, the physics behind it, and every control

At a glance

Eight planets move under a single force, the Sun's gravity. This sim lets you follow them round their orbits and read each planet's distance from the Sun and its speed, so you can see what Kepler's laws describe.

What you'll learn

How a planet's speed changes as it goes round its orbit.
Where the Sun sits relative to an orbit, and how the planets' orbit shapes differ.
How the size of an orbit is linked to a planet's speed and to the length of its year.
How one force, gravity, accounts for all of these.

r: distance from the Sun. v: the planet's speed, along its path.

What's on screen

Sun: the bright disc at the centre of the view.
Planets: coloured dots on their orbit rings. Distances are to scale; the dots are enlarged to stay visible.
Selected planet panel: the chosen planet's name, a, e, and its current distance r and speed v.

See itSet the planets moving at 1× from the start. Watch for: the planets going round their rings, each at its own pace.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Zoom	Inner shows Mercury to Mars at true scale. Full shows all eight planets, with orbit sizes compressed so Neptune fits; the shapes are unchanged.	Inner / Full opens on Inner
Highlight planet	Choose one of the eight planets. Its ring brightens and the Selected planet panel switches to it. Click it again to clear.	8 planets none at the start
Selected planet	Readouts for the chosen planet: a (semi-major axis, AU), e (eccentricity), r (distance now, AU) and v (speed now, km/s).	panel
The orbital view	Drag to tilt and turn the camera. Scroll to zoom. This changes only your viewpoint, not the motion.	drag · scroll
▶ Play / ⏸ Pause	Starts the planets round their orbits, or freezes them so you can read the panel.	top bar
Time rate	How fast simulated time runs. At 1×, one second on screen is 0.02 years (about 7 days). The orbits themselves do not change.	0.25×–8× opens at 0.25×
↻ Reset	Puts every planet back at the start of its orbit and clears the selection.	top bar

Fixed: each planet moves only under the Sun's gravity. The Sun stays still, the planets do not pull on each other, and every planet starts at its closest point to the Sun. While the Guided Inquiry runs, some readouts unlock step by step.

The Sun sits at a focus

The Sun's pull on a planet points at the Sun and weakens as 1/r². In AU and years, GM = 4π².

This force alone makes the path an ellipse with the Sun at one focus (Kepler's first law). Here a is the semi-major axis, and e is the eccentricity, with e = 0 a circle. θ is measured from the closest point.

Mercury's orbit (e = 0.206) beside a circle with the Sun at its centre.

The Sun sits a·e from the centre of the ellipse. For Mercury that is 0.08 AU, about a quarter of its closest distance to the Sun (0.31 AU).
The ring itself is only slightly squashed: its short axis is √(1 − e²) = 0.979 of the long one. The off-centre Sun is the clearer sign.
The sim's camera looks at the orbits at an angle, which squashes every ring on screen. That is perspective, not eccentricity.

See itSelect Mercury in the Inner view. Watch for: the Sun sitting well to one side of the middle of Mercury’s ring, with e = 0.206 in the panel.

Planets run fastest near the Sun

Gravity pulls straight toward the Sun, so it exerts no torque about the Sun. The planet's angular momentum L stays constant, and the line from the Sun to the planet sweeps out equal areas in equal times (Kepler's second law).

Mercury at e = 0.206. Each shaded sector is swept in 7.3 days (T/12).

At the closest and farthest points the velocity is perpendicular to r, so r·v is the same at both: v_peri / v_apo = r_apo / r_peri = (1 + e) / (1 − e).
Mercury: 58.98 km/s at 0.31 AU and 38.86 km/s at 0.47 AU, a ratio of 1.52. Earth (e = 0.017): 30.29 and 29.29 km/s.
Energy tells the same story: falling toward the Sun turns potential energy into kinetic energy.

See itRun Mercury from its closest point at 2×. Watch for: v (now) largest when r (now) is smallest, and both reversing on the far side.

See itRace Mercury against a ghost on a ring of radius a, circling at a steady speed with the same period. Watch for: Mercury pulling ahead of the ghost, then the ghost catching up at the far end of the orbit.

Speed follows from r and a

Energy conservation gives the speed at any point of an orbit from just r and a:

v comes out in AU/yr. Multiply by 4.74 for km/s.

Mass and direction of travel do not appear. Only where the planet is (r) and which orbit it is on (a).
At r = a the speed is √(GM/a), the speed of a circular orbit of that radius. Closer in, the planet is faster; farther out, slower.

Planet

This slider selects the planet in the sim too.

See itRun the planet chosen above from its closest point. Watch for: v (now) swinging between the two predicted values as r (now) swings between its predicted values.

Bigger orbits mean slower planets

Across the family, a sets both the period T and the speed (Kepler's third law, in years, AU and km/s):

All eight planets: speed against orbit size on log axes.

Earth: a = 1 AU, T = 1 yr, v ≈ 29.8 km/s. Neptune: a = 30.06 AU, so T = 30.06^3/2 = 164.8 yr and v ≈ 5.4 km/s.
A planet 30 times farther out is √30 ≈ 5.5 times slower, not 30 times. The Sun's pull is weaker there, so less speed holds the orbit.
Both relations follow from gravity: on a near-circular orbit v²/r = GM/r², so v = √(GM/r) and T = 2πr/v.

The panel shows v at the moment, so it matches 29.8/√a only where r ≈ a.

See itRead a and v for Earth, then for Neptune. Watch for: a about 30 times larger for Neptune, v about 5.5 times smaller.

Where it matters

Satellites. Any orbit obeys the same laws. A circular, equatorial orbit of radius 42,164 km around Earth has a period of 23.93 h, one turn of the Earth, so the satellite stays over one spot (geostationary orbit).
Weighing the Sun. T² = 4π²a³/GM turns a measured a and T into a mass. Earth's orbit gives M_☉ ≈ 1.99 × 10³⁰ kg, and a moon's orbit gives its planet's mass the same way.
Comets. Halley's comet (e ≈ 0.97, a ≈ 17.8 AU) passes within about 0.59 AU of the Sun and recedes to about 35 AU. By r·v conservation it is about 60 times faster at perihelion than at aphelion.

Go deeper: where the speed formula and GM = 4π² come from

Energy per unit mass is ½v² − GM/r = −GM/(2a). Solving for v² gives v² = GM(2/r − 1/a). The energy is negative for every bound orbit.
Written with GM, Kepler's third law is T² = 4π²a³/GM. Choosing GM = 4π² AU³/yr² makes it T² = a³.

Summary

In one line. The Sun's gravity alone puts each planet on an ellipse with the Sun at a focus. The planet is fastest where it is closest, and the larger the orbit, the slower the planet and the longer its year.

Key points

The Sun is at a focus: closest distance a(1 − e), farthest a(1 + e).
r·v is the same at both ends, so speed rises as distance falls. The full law is v² = GM(2/r − 1/a).
T² = a³, and v ≈ 29.8/√a km/s on a near-circular orbit.

Common mix-ups

"The Sun is at the centre of each orbit." It sits at a focus, a·e from the centre.
"A planet moves at a constant speed." The speed changes. What stays constant is the area swept per unit time.
"A planet 30 times farther out is 30 times slower." The speed scales as 1/√a, so it is about 5.5 times slower.

Things to try on your own

Select Mercury at 2×. Note r and v at their smallest and largest, and check that r × v matches.
In the Full view, read a and v for Venus, Earth and Neptune. Check that v × √a is about 30 for each.
Set Time rate to 8× and time one Earth lap. Expect 1 yr ÷ (0.02 yr/s × 8) = 6.25 s.