Elliptical Orbit

ORBITREK MX - Seated Under Desk Elliptical Machine, Low Impact, Compact Bike Pedal Exerciser Machine with 8 Exercise Options. A Machine to Experience “Active Sitting” for a Healthier, Better Life! In orbit of another planet, elliptical; some elliptical orbits are very nearly circles, while others are much elongated. Some bodies may follow parabolic or hyperbolic paths (open-ended curves). The orbit of a body approaching the solar system from a very great distance, curving once around the Sun, and receding again is such Read More.

So far, we've examined the methods by which several propertiesof stars can be measured:distance, luminosity, temperature, size.Another fundamental property of a star, or any celestial object,is its mass.How can one measure the mass of a star?It turns out that one must find a star whichis in orbit around another star(s)and use gravity as a tool to turn orbital motion into mass.

But it's a complicated subject.Let's start off with a simple example of orbital motion --planets in our own solar system -- and work our way up to the more complex cases of distant stars.

Orbits are usually elliptical, this demonstrates an easy way to draw elliptical orbits that could represent orbits of planets, asteroids and comets in the so.

Kepler's First Law: shape of the orbit

Johannes Kepler was a brilliant mathematician who livedin the late sixteenth and early seventeenth century,a contemporary of Tycho Brahe, Galileo, and Queen Elizabeth I.In order to gain access to the best measurementsof planetary motions in the world, he became Tycho Brahe'sassistant; his job was not to make the observations,but to analyze them.After Tycho's death in 1601, Kepler spent yearsworking on the great mass of measurements Tycho had acquired.

In 1609, Kepler published a book, Nova Astronomica,in which he made a revolutionary claim:planets don't move in CIRCLES, as everyone had previously thought(including Copernicus);instead,

Kepler's First Law: the planets move in ellipses, with the Sun at one focus.

Let's pause to consider the properties of an ellipse.There are several ways to define this shape:

The locus of all points which lie a fixed sum of distances away from two foci.
Stick two pins into a piece of paper, and place a loop of string loosely around the pins. Now hold a pen inside the loop and pull outwards until it stretches the string taut. Move the pen around the pins, always keeping the string stretched tightly.
In a cartesian coordinate system with axes x and y,
In a polar coordinate system,

Below is a sample ellipse drawn on a background grid.Mark on your piece of paper the following quantities;make all measurements in units of the grid spacing.

Measure the semimajor axis a
Measure the semiminor axis b
Calculate the eccentricity e using the formula
Calculate the positions of the two foci. Each one lies along the semimajor axis, a distance ae from the center of the ellipse.
Mark the two foci. Call the right-hand focus the 'principal focus,' and draw the Sun at that position.
Calculate the perihelion distance and the aphelion distance.

Kepler's Second Law: motion around the orbit

Using Tycho's careful measurements of the position of Marsin our sky, Kepler was able to find a second'imperfection' in the motions of the planets.Not only did they move in ellipses (instead of circles),but their speed was not constant!Instead, the planets move quickly when close to theSun and slowly when far from the Sun.A nice geometric description is

Kepler's Second Law:A line connecting a planet to the Sun sweeps outequal areas in equal times.

Use this rule on your paper:

The planet moves exactly 4 grid units along its orbit at aphelion during time period T. Mark the starting and ending points of this interval on the orbit. Draw the triangle which describes the area swept out during this interval (the apex of the triangle should be the Sun).
What is the area of this triangle?
Consider a time interval T centered on the planet's passage across the semiminor axis of the orbit. How far along its orbit will the planet move during this interval? Mark the starting and ending points of this interval, and draw the triangle swept out.
Consider a time interval T centered on the planet's passage through perihelion. How far along its orbit will the planet move during this interval? Mark the starting and ending points of this interval, and draw the triangle swept out. (Approximate values are okay).
What is the ratio of orbital speed at perihelion compared to orbital speed at aphelion for this orbit?

Note that Kepler's Second Law is nice, but it deals onlywith RELATIVE speeds.We can't use it directly to figure out exactly how fara planet will move over a period of, say, three weeks.Is there any way to figure out exactly wherea planet will be at some particular time?

Yes -- two ways, in fact:

classical approach -- Kepler's equation
modern approach -- brute-force numerical integration

Thanks to fast, cheap computers, we members of the modernworld can simply plug some initial conditions intoa computer program and integrate the motion of the bodiesnumerically to follow their motion as a function of time.One of the problems on this week's homework asks you to dojust that.

Back in the old days, however, computing wasn't quite so cheapand easy. Kepler didn't have computers or calculators.In fact, Kepler didn't even have LOGARITHMS; Napierpublished his first tables in 1632, after Kepler hadderived his laws of planetary motion.Kepler did manage to devise a method of computing the position of a planet at any time.It involved a bit of calculating, but nowhere nearas much as the modern brute-force approach.Since it's an interesting little mathematical puzzle,let's look at it in some detail.

Kepler's equation for motion around an orbit

The problem is this:we know the orbital parameters of a planet's motion around the Sun:period P, semimajor axis a, eccentricity e.We also know the time T when the planet reachesits perihelion passage.Where will the planet be in its orbit at some later time t?

If the orbit is circular, then this is easy:the fraction of a complete orbit is equal tothe fraction of a complete period which has elapsedsince the last perihelion passage.For example, if (t - T) is exactly one-quarter of theperiod P, then the planet will have madeexactly one-quarter of a full circle around the Sun.

But, as Kepler's Second Law states, planets in ellipticalorbits do NOT move with a constant speed, nor with aconstant angular speed. In real life, some planetaryorbits are significantly non-circular, so the circularapproximation won't work.What can we do?

We can measure the position of a planet in its ellipticalorbit with the angle between its radius vector and the perihelion position.This angle is called the true anomaly,and is conventionally written as the letter v.

Yes, I've moved the principal focus closer to the center of the circle than it should be, for clarity.

What is v as a function of time t?

Kepler found an answer to this question, but it requireda bit of a roundabout journey.He discovered that if he defined a 'auxiliary' quantity,he could solve for that quantity as a function of time;and then he could convert from the 'auxiliary' quantityto the desired true anomaly v.

Kepler's first step was to draw a circle around the ellipse,and project the position of the planet on its ellipticalorbit upwards to meet the circle.

The angle E measured from perihelion position, to centerof circle, to projected position of planet, is calledthe eccentric anomaly.If one can find E, one can go back to thedesired variables r and v like so:

Kepler's next step was to find a mathematical relationshipbetween this eccentric anomaly E and time.He computed n, the average angular speed of the planet,also called the mean motion,

You should express the mean motion, and all angularquantities involved in Kepler's equation,in radians.

Finally, he was able to write the following,known as Kepler's equation:

Looks good, right? You have to go through several steps:

find the time of perihelion passage T
calculate the mean motion n
solve Kepler's equation for the eccentric anomaly E
convert to the true anomaly v and radius vector r

but it's not so bad, is it?

Actually, it is. Step 3, solving Kepler's equation for Egiven some time difference (t - T),turns out to be a doozy.here is no simple closed-form solution, in general,though there are series solutions which converge quicklyfor small values of the eccentricity.AstronomerFriedrich Bessel devised his eponymous 'Bessel Functions'in 1817 as a means to solve Kepler's equation.Letting M = (t - T) n (this is known as the 'mean anomaly')one can write

where Jk is the k'th Bessel function of the first kind.

Fortunately, even though there isn't a closed solution,Kepler's equation yields pretty quickly to any numberof numerical attacks, especially when the eccentricity e is small.You can program a computer to use Newton's method or the the bisection technique. You can even fall back on trial and error.Give it a try.

Exercise: Your piece of paper shows an elliptical orbit. Suppose that the orbit is P = 500 days

in a counterclockwise direction. Where will the planet be at (t - T) = 400 days after perihelion passage? Calculate the true anomaly angle v

Elliptical Orbit Example

and use it to mark the position of the planet along the orbit.

Elliptical Orbit Theory

Start by finding the mean motion n and the mean anomaly M = n(t - T). Use a starting guess that the eccentric anomaly E is equal to the mean anomaly. Plug that E into the left-hand side of Kepler's equation and see what you get. If you get exactly M, you're done! Otherwise, modify your value of E and try again. See if you can find E to four significant figures.

Elliptical Orbit Define

For more information

Laurence G. Taff's book Celestial Mechanics provides more information than you'd ever want to know about all kinds of ways to describe orbits and then solve for their motions.

Learn about this topic in these articles:

comets

In comet: Ancient Greece to the 19th century
…Any less-eccentric orbits are closed ellipses, which means a comet would return.
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In comet: Ancient Greece to the 19th century
…orbit was indeed a closed ellipse. Moreover, he showed that the orbital period of the comet around the Sun was only 3.3 years, still the shortest orbital period of any comet on record. Encke also showed that the same comet had been observed by French astronomer Pierre Méchain in 1786,…
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In comet: Ancient Greece to the 19th century
…that most comets were on elliptical orbits and thus were members of the solar system. Many were recognized to be periodic. But some orbit solutions for long-period comets suggested that they were slightly hyperbolic, suggesting that they came from interstellar space. That problem would not be solved until the 20th…
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In comet: The modern era
…the apparently hyperbolic orbits became elliptical. That proved that the comets were members of the solar system. Orbits of that type are referred to as “original” orbits, whereas the orbit of a comet as it passes through the planetary region is called the “osculating” (or “instantaneous”) orbit, and the orbit…
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In comet: The modern era
…sufficient to change the previously elliptical orbits of the comets to hyperbolic, ejecting them from the solar system and sending them into interstellar space. Van Woerkom also showed that because of Jupiter, repeated passages of comets through the solar system would lead to a uniform distribution in orbital energy for…
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In comet: The modern era
…often changed from hyperbolic to elliptical. Very few comets were left with hyperbolic original orbits, and all of those were only slightly hyperbolic. Marsden had provided further proof that all long-period comets were members of the solar system.
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Kepler’s laws

In Kepler's laws of planetary motion
…move about the Sun in elliptical orbits, having the Sun as one of the foci. (2) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time. (3) The squares of the sidereal periods (of revolution) of the planets are directly proportional to…
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In celestial mechanics: Kepler’s laws of planetary motion
An ellipse (Figure 1) is a plane curve defined such that the sum of the distances from any point G on the ellipse to two fixed points (S and S′ in Figure 1) is constant. The two points S and S′ are called foci, and the…
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planetary orbits

Elliptical Orbit Of Earth

In orbit
…of another planet, elliptical; some elliptical orbits are very nearly circles, while others are much elongated. Some bodies may follow parabolic or hyperbolic paths (open-ended curves). The orbit of a body approaching the solar system from a very great distance, curving once around the Sun, and receding again is such…
Read More
In solar system: Orbits
…move around the Sun in elliptical orbits in the same direction that the Sun rotates. This motion is termed prograde, or direct, motion. Looking down on the system from a vantage point above Earth’s North Pole, an observer would find that all these orbital motions are in a counterclockwise direction.…
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