Elliptical Orbits.
The conical sections – the circle, the ellipse, the parabola and the hyperbola – were all known in antiquity, more than 2000 years ago. Many properties of ellipses had already been worked out in detail, and were familiar to mathematicians in the renaissance. In 1609 the German mathematician and astronomer Johannes Kepler discovered that all planets orbit the sun in elliptical paths. He painstakingly derived that from the excellent data collected by his mentor, the Danish astronomer Tycho Brahe. His conclusions also known as Kepler’s Laws:
- Planets orbit the sun in elliptical orbits while the sun is located. off center, in the focus of the orbit.
- The speed of a planet varies as it orbits the sun. The imaginary line connecting the sun and the planet sweeps out equal areas in equal time intervals.
- The time period for a revolution squared is directly proportional to the cube of the semi major axis
Kepler’s conclusions were purely based on geometrical observations. He had no idea about why this was so. These laws however were necessary stepping stones for Isaac Newton to develop the theory of gravitation. Newton published his theory in 1687, almost 80 years after Kepler’s discoveries.
Here are some definitions and basic formulas for an ellipse as shown above.
- “a” is the semi major axis, “b” is the semi minor axis, “c” is the focal distance.
- an ellipse can be drawn with a string connected to two focal points where the string is larger than 2c. The resulting ellipse major axis (2 times “a”) is equal to the length of the string (PF1+PF2).
- The eccentricity of an ellipse is the focal distance divided by the semi major axis or e=c/a.
- The area of an ellipse is pi times the semi major axis times the semi minor axis.
\begin {aligned} Ellipse Area=\pi. a.b \end{aligned}- The square of semi major axis is equal to the sum of the squares of the focal distance and the semi minor axis.
\begin {aligned}a^2=(b^2+c^2 ) \ frequently \ written \ as:\ c=\sqrt{a^2-b^2}\\\end{aligned}Kepler’s second law states that the imaginary line between the sun and a planet sweeps out equal areas in equal time periods. This means that the ratio between the area and the time interval is constant. The area swept out at any given small time interval is equal to the total area of the ellipse divided by the total orbit time. This has an interesting implication for an easy derivation of the speed at aphelion (furthest from the sun) and the speed at perihelion (nearest to the sun). At these two points in the planets orbit, the speed vector is perpendicular to the line between the sun and the planet. Actually, if you have both the aphelion and perihelion distances you could use the circular orbit formulas from the previous page to get a close approximation of the elliptical orbital speed at these points.
In any case, here is the derivation of the speed formula for aphelion and perihelion using the ellipse geometry of the earth orbit.
As the earth nears aphelion, the imaginary line R sweeps out an area given by the formula
\begin {aligned}Area=\frac12Base. v.\Delta t \ At \ aphelion \ the \ infinitesimal \ area \ would \ be \ \frac12.R.v.dt\end{aligned}The area of a right angle triangle is 1/2 times the product of the base and the height. As the Radius R approaches Perihelion, the base approaches the length of R. Even at this infinitesimal distance the ratio of the Area divided by the the time interval ( dt) is the same as the ratio of the whole orbit area divided by the period of a complete orbit.
\begin {aligned}\frac12.R.v.\frac{dt}{dt}=\frac{\pi.a.b}{P} \ This \ equation \ solved\ for\ velocity
\ v\ is: v=\frac{2.\pi.a.b}{R.P} \end{aligned}Here are the orbital data from the NASA Earth Fact Sheet:
Semimajor axis (106 km) 149.598 Sidereal orbit period (days) 365.256 Tropical orbit period (days) 365.242 Perihelion (106 km) 147.095 Aphelion (106 km) 152.100 Mean orbital velocity (km/s) 29.78 Max. orbital velocity (km/s) 30.29 Min. orbital velocity (km/s) 29.29 Orbit inclination (deg) 0.000 Orbit eccentricity 0.0167 Sidereal rotation period (hrs) 23.9345 Length of day (hrs) 24.0000 Obliquity to orbit (deg) 23.44 Inclination of equator (deg) 23.44
As you can see “b” the semi major axis is missing in the table above. However the orbit eccentricity “e” is available. “b” can then be calculated from the following formula.
\begin{aligned} a^2=b^2+c^2 \ or \ b=\sqrt(a^2-c^2)\end{aligned}\begin{aligned}\ eccentricity\ is\ defined\ as: e=\frac{c}{a}\ or \ c=e.a\end{aligned}\begin{aligned}\ The\ semi\ minor\ axis\ can\ then\ be\ calculated: b=\sqrt(a^2-e^2a^2)\end{aligned}With the proper substitutions the formula for the velocity v at aphelion or perihelion is:
\begin{aligned}v=\frac{2.\pi.a.\sqrt(a^2-e^2a^2)}{RP}=\frac{2.\pi.a^2.\sqrt(1-e^2)}{RP} \end{aligned}The result after adjusting the values for meters and seconds, and plugging in the distance to the sun at aphelion the calculation result is 29291 m/sec or 29.291 km/sec which is very close to the NASA fact sheet for minimum orbital speed.
At perihelion plug in the shorter distance for to the sun for R, and the values given in the NASA fact sheet. Be careful to convert the values to meters and seconds. After adjusting the result from meters to kilometers you get 30.287 km/sec again close enough to the value in the NASA table.
The circular orbits from the previous page give results for R=147095000 km and R=152100000 km. Respectively 30.04 km/sec and 29.54 km/sec. That is within 1% of the published values.