Every planet in the solar system moves by three rules that a German astronomer wrote down four centuries ago, long before anyone knew why they worked. They still hold up well enough to plan a trip to Mars on the back of an envelope.
The new Orbit Lab lets you see those rules rather than just read them. It has three modes: Kepler’s laws on an orbit you design, the eight planets on real orbital data, and a Hohmann transfer planner for travelling between any two of them.
Kepler’s three laws
Johannes Kepler published the first two in Astronomia nova in 1609, although the second law reached its modern form in his Epitome of Copernican Astronomy in 1621. The third appeared in Harmonice Mundi in 1619.
First law. Each planet’s orbit is an ellipse with the Sun at one focus, not at the centre. Second law. A line from the Sun to a planet sweeps out equal areas in equal times. Third law. The square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit.
In the first mode, drag the eccentricity slider up and watch the circle stretch into a long ellipse. Leave the “Equal-area slices” button on and the app shades the path in time slices of equal length. Each slice has the same area, so near the Sun it is short and fat while far away it is long and thin. That geometry is the reason a planet speeds up as it dives toward the Sun. The side panel gives the speed at perihelion and aphelion from the vis-viva equation, using the Sun’s gravitational parameter.
For a small body orbiting the Sun, the third law is simple when you measure in years and astronomical units: the period in years equals the semi-major axis in AU raised to the power 1.5. Set the semi-major axis to 4 AU and the period reads 8 years.
The real planets
Switch to Solar system mode and the app places Mercury through Neptune on their actual orbits. The inputs are the distance from the Sun, the orbital eccentricity and the orbital period from the NASA Planetary Fact Sheet. Select any planet to read its distance and speed right now, plus a check of T²/a³. In years and AU, Kepler’s third law says that should be 1. It lands between 0.996 and 1.001 for every planet except Saturn, at 0.987, and Neptune, at 0.975. I would treat those two as a sign that NASA’s “distance from Sun” column is not exactly the semi-major axis for the outer planets, rather than as a failure of the law, but I have not checked that explanation.
Two things stand out. Most orbits look almost circular because their eccentricities are small: Venus is 0.007 and Neptune 0.010. Mercury is the exception at 0.206, and Mars at 0.094 is the noticeable one among the inner planets. And the log distance scale button is worth a click, because on a linear scale the inner solar system is crushed into a dot beside Neptune.
Open Orbit Lab in its own tab, or use its Full screen button.
Going to Mars: the Hohmann transfer
Walter Hohmann described the cheapest two-burn route between circular, coplanar orbits in his 1925 book Die Erreichbarkeit der Himmelskörper. The craft leaves the inner orbit on an ellipse whose near point touches that orbit and whose far point touches the outer one. A burn at the start puts the craft on the ellipse, and a second burn at the far end circularises it.
Pick Earth and Mars in the third mode and the app computes the rest. The transfer takes about 259 days. For the craft to meet Mars at the far end, Mars must start about 44 degrees ahead of Earth at departure, which is why the animation sets them up that way. The two heliocentric burns come out near 2.95 and 2.65 km/s. Wikipedia describes the Earth-to-Mars trip as about nine months, with launch windows every 26 months, and my own arithmetic from the planets’ periods lands on a gap of roughly 780 days between windows, which matches.
Do not read those burn numbers as a mission budget. They are changes in speed relative to the Sun and they leave out the work of climbing out of Earth’s gravity well. Wikipedia quotes about 3.6 km/s for the departure burn from low Earth orbit, and the arrival burn depends on how the craft is captured. Real missions also fly orbits that are slightly tilted, with planets on ellipses rather than circles.
What the app leaves out
All planets sit in one plane and start at the same angle at time zero, so the app shows the pattern of orbits rather than a calendar of where the planets are today. Planet masses never tug on each other here, nor does the Sun wobble around the system’s centre of mass. Those effects are small but real, and they are what separates this model from a proper ephemeris. For gravity that does include planets pulling on each other, see the Gravity Lab, and for what happens when a three-body system turns unpredictable, the chaos attractors post.
Sources
- NASA Planetary Fact Sheet (distance from Sun, orbital period, eccentricity): nssdc.gsfc.nasa.gov/planetary/factsheet
- Kepler’s laws of planetary motion, Wikipedia (publication dates, statements): en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
- Hohmann transfer orbit, Wikipedia (1925 book, Earth–Mars time, 3.6 km/s, 26-month windows): en.wikipedia.org/wiki/Hohmann_transfer_orbit
