Elliptical orbits and the vis-viva equation

Talking points

• NASA (Orbits and Kepler's Laws): each planet's orbit is an ellipse with the Sun's centre at one focus, so the planet-to-Sun distance changes all the way round.
• NASA Mercury Fact Sheet: perihelion 46.000, aphelion 69.818 and semi-major axis 57.909 million km; eccentricity 0.2056. The Planetary Fact Sheet lists 0.206 for Mercury against 0.094 for Mars and 0.017 for Earth, the largest of the eight planets.
• JPL Horizons (DE441) puts this perihelion at 08:28 UT on 10 November 2026. The viewer draws the planets from JPL's approximate Keplerian elements, whose nominal error for Mercury between 1800 and 2050 is 15 arcseconds in longitude; its own perihelion falls at 08:31 UT.
• This is true scale. Mercury is far smaller than a pixel from here, so the label marks it. The ellipse is visibly off-centre: the Sun sits 0.08 au from the middle of it.

Talking points

• Derivation: perihelion q = a(1 − e) and aphelion Q = a(1 + e), so a = (q + Q)/2 and e = (Q − q)/(Q + q).
• NASA's fact sheet notes give the semi-major axis as the average distance, midway between perihelion and aphelion. NASA's Basics of Space Flight defines eccentricity as the distance between the foci divided by the major axis: the foci are Q − q = 2ae apart and the major axis is Q + q = 2a.
• Mercury: a = (46.000 + 69.818)/2 = 57.909 million km, the fact sheet's value. e = 23.818/115.818 = 0.20565, which agrees with the fact sheet's 0.2056 to the rounding of its distances (its J2000 mean elements give 0.20563).
• The Sun is ae = 11.909 million km (0.080 au) from the centre of the ellipse.
• JPL Horizons: aphelion on 24 December 2026 at 08:06 UT, 69.817 million km from the Sun.

Talking points

• At the apsides the velocity is at right angles to the radius, so conserving angular momentum per unit mass gives q·v_p = Q·v_a.
• The energy per unit mass, v²/2 − GM/r, is the same at both apsides. Put v_a = v_p·q/Q into it: (v_p²/2)(1 − q²/Q²) = GM(1/q − 1/Q), so v_p² = 2GM·Q/(q(q + Q)) = GM(2/q − 1/a), using q + Q = 2a.
• Then the energy is v_p²/2 − GM/q = −GM/(2a), which depends on a alone. Setting v²/2 − GM/r = −GM/(2a) at any point of the orbit gives v² = GM(2/r − 1/a).
• JPL gives the Sun's gravitational parameter as GM☉ = 1.32712440041 × 10²⁰ m³ s⁻² (from the DE440 ephemeris). Mercury's orbital energy is then −GM☉/(2a) = −1.1459 × 10⁹ J per kilogram.
• Aphelion by the same route: 2/Q − 1/a = 2.864591 × 10⁻¹¹ − 1.726847 × 10⁻¹¹ = 1.137744 × 10⁻¹¹ m⁻¹, so v_a = 38.86 km/s. NASA's minimum orbital velocity is 38.86 km/s.

Talking points

• NASA: the imaginary line joining a planet and the Sun sweeps equal areas in equal times, so a planet moves fastest at perihelion and slowest at aphelion.
• The areal rate is half the angular momentum per unit mass, so equal areas is the same statement as q·v_p = Q·v_a. Then v_p/v_a = Q/q = 69.818/46.000 = 1.518; NASA's maximum and minimum speeds give 58.97/38.86 = 1.517.
• The angular speed at an apsis is v/r, so its ratio is (Q/q)² = 2.30: Mercury turns 6.35 degrees a day round the Sun at perihelion and 2.76 degrees a day at aphelion, against a mean of 360/87.969 = 4.09.
• The half of the orbit nearer the Sun (true anomaly from −90 to +90 degrees) takes 32.5 days and the far half 55.4 days. At true anomaly 90 degrees the eccentric anomaly is E = arccos e = 78.13 degrees and the mean anomaly M = E − e·sin E = 66.60 degrees, so the near half takes 2 × 66.60/360 = 0.370 of the period.
• The same equation works for Earth: q = 147.095 and Q = 152.100 million km give 30.29 and 29.29 km/s, NASA's maximum and minimum for Earth.

Talking points

• The clock opens at JPL Horizons' perihelion, 10 November 2026 08:28 UT. Aphelion is on 24 December 2026 at 08:06 UT.
• There is no check on Mercury's distance, so this one reads the date through the Sun's longitude of date: it passes within 3 degrees of 272.5 degrees, the value at Mercury's aphelion. The Sun moves 1.02 degrees a day in late December, so that is about three days either side, and a pause inside it gives 41 to 47 days.

Talking points

• NASA: the squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits.
• For a circular orbit, equating gravity to the centripetal acceleration, GM/a² = 4π²a/P², gives P = 2π√(a³/GM) directly. Kepler's third law says the same holds for an ellipse with a the semi-major axis.
• The law strictly uses G(M☉ + m). JPL gives Mercury's GM as 22,031.87 km³ s⁻², 1.66 × 10⁻⁷ of the Sun's, so leaving it out changes the period by under one part in ten million.

Talking points

• At r = a vis-viva gives v² = GM(2/a − 1/a) = GM/a, so v = √(1.32712 × 10²⁰ / 5.7909 × 10¹⁰) = 47.87 km/s.
• r = a(1 − e·cos E) equals a where cos E = 0, at the two ends of the minor axis, a quarter of the way round in eccentric anomaly. That is M = 90 − e × 57.296 = 78.22 degrees of mean anomaly, 19.11 days after perihelion. JPL Horizons has Mercury crossing r = a on 29 November 2026 between 11:00 and 12:00 UT, and the scene opens at 11:30.
• NASA's fact sheet notes describe the orbital velocity as the average speed of the planet as it orbits the Sun. Averaged over time, that is the length of the orbit divided by the period: with b = a√(1 − e²) = 56.671 million km, Ramanujan's approximation for the perimeter, π[3(a + b) − √((3a + b)(a + 3b))], gives 359.97 million km, and 359.97 × 10⁶ km / 7.6006 × 10⁶ s = 47.36 km/s.