The Kepler Equation
from Gerhard Holtkamp, 24. June 2009, 21:47
An equation which was first published 400 years ago continues to be in demand - more than ever before.
This year has been designated the International Year of Astonomy as we celebrate the 400th anniversary of two seminal events: Galileo's telescopic observations of the sky and the publication of the Astronomia Nova by Johannes Kepler.
In his book Kepler describes in detail how he first found what we now call Kepler's Second Law (a line drawn from the Sun to the moving planet will sweep equal areas in equal times - actually a special case of the law of conservation of angular momentum) and toward the end of the book the discovery of Kepler's First Law (which states that the planets are orbiting the Sun on ellipses rather than circles). Kepler managed to achive this task by a combination of sound mathematical and astronomical knowledge, access to the most accurate observational data available at the time, hard work with a lot of trial and error but also by brilliant intuition.
In reviews of popular science books you often read that the author managed to tell everything without using a single mathematical formula. Well, the Astronomia Nova should get high marks then. It is packed with mathematics but there is not a single formula to be found! How then does Kepler explain all the mathematics? By using lots of diagrams and even more text. As an example here is a line from chapter 47 (out of 70 chapters):
"You now have to determine the angle Bβδ in such a way that the product of its sine BC and half of αβ, which is to say the area of the triangle αBβ together with the sector Bβδ is equal to the area which before has been found from the time."
This sentence refers to a diagram (which I have greatly simplified - the original diagram contains many more lines and letters that are needed all throughout this chapter).
Actually, the quote you just read and most likely didn't understand is nothing less than the first mentioning of what has since become known as the Kepler Equation. In modern textbooks it looks like this:
M = E - e sin(E)
Here you can see the beauty and power of modern mathematical notations. What has been a difficult to understand sentence in the Astonomia Nova is now condensed into a simple formula. So what does Kepler's equation tell us?
Let's start with M. This is called the Mean Anomaly and is an angle which increases linearly with time just like the hour hand of a clock would move once around in 60 minutes. If we know M we know the corresponding time and vice versa.
The e in front of the sine function is called eccentricity. It is 0 for a perfect circle and the larger it gets the more elongated the ellipse will be until after passing a value of 1 it is no longer a closed curve and turns into an open hyperbola. This is the trajectory of many comets and also of spacecraft leaving Earth on an interplanetary trajectory.
E is called the Eccentric Anomaly. This is an auxiliary angle which is directly related to yet another angle called the True Anomaly ν. It is this final angle which gives you the position of the planet along the ellipse. So knowing E is as good as knowing the position of the planet. If you know the eccentricity (which is one of the parameters of the orbit) and you have a position E then Kepler's equation will give you M and thus the corresponding time right away.
But usually you are interested in the opposite problem. You want to find the planet's position at a given time. It is now that our innocent looking little equation suddenly turns nasty. We know M and want to find E. E however appears not just by itself but also as an argument of the sine function. We cannot resolve Kepler's equation for E and get a straight answer. So what to do? Well you can always guess a value and see whether it fits and if it doesn't try another one. Mathematicians have found ways to do this systematically. Kepler knew of the Regula Falsi which you may or may not have heard about in school. But we have more efficient methods today. It typically takes just a few iterations to come up with a sufficiently precise answer to Kepler's equation.
In order to find a planet's position you need some information about it's orbit. We already learned of the eccentricity describing the shape of the ellipse. The size of the ellipse is given by the semi-major axis which in case of a circle would correspond to its radius. Three angles are needed to specify how the ellipse is oriented in space and finally we have to know where the planet was at a specific reference time. This set of data is usually referred to as the Kepler elements of an orbit.
Sophisticated methods exist today which generations of mathematicians have perfected to find the orbit elements from observed positions of the planet. How did Kepler do it? By having a clear three-dimensional understanding of the orbit in space he realized that certain privileged positions of the planets would help him find these orbit elements. He also had a good feeling about which observations to trust more than others. All this makes very interesting reading in the Astronomia Nova although most people will find this book slightly less easy to read than a paperback novel.
Once you know the Kepler elements of an orbit you can calculate the position of a planet via the Kepler equation for any time in the future or past. Well - not quite. More than one planet (and other objects) are orbiting the Sun and they all tug on each other via gravitation. There are also relativistic effects. This distorts our neat ellipses a little. For precise orbit calculations we now use methods of numerical integration.
The drawback of numerical methods is that you have to take one small step at a time and so it takes many time steps (and associated calculations) to finally arrive at the point in the future which you are interested in. By contrast you need just one calculation of the Kepler equation to find the position of the planet directly at any one time. Because this is a lot faster Keper's equation is usually employed for calculating the orbits of some tens of thousands of minor planets which zip around the solar system. Here Kepler elements are used which will give sufficiently accurate results for a few years after which they have to be updated.
Satellites orbiting the Earth are subject to a number of forces. The gravitational field of the Earth is not symmetric and has many bumps which influence the orbit. Satellites in low orbits are affected by atmospheric drag. Light pressure, Sun, Moon and relativity are also distorting our Kepler ellipses. Once again for high precision work you use numerical integration. But to do that you need accurate orbit information which you normally have for your own satellites but only rarely for others.
Most orbit information publicly available is in the form of so called Two-Line-Elements (TLE). This is essentially a set of Kepler elements with one extra number added to account for atmospheric drag. The procedure to calculate a satellite's position from TLEs runs over several pages and most of it is to account for the perturbing forces. But toward the end a few lines appear with a comment that here Kepler's equation is being solved.
Calculating satellite positions via TLEs is used by observers who want to know when to see the ISS or other satellites in the sky. But there are many other uses where the accuracy of this analytic method is adaquate. By far the biggest customers for TLEs (and thus for Kepler's equation) are the people who deal with space debris.
In addition to about 900 active satellites there are many other objects in orbit - inactive satellites, spent rocket stages, pieces left from explosions etc.
TLEs for more than 13000 objects are publicly available on the Internet (some 4000 more objects are individually tracked but are not publicly listed). In order to avoid a collision between an active satellite and some space debris you must check all the orbits a few days in advance and if there will be a close encounter perform an avoidance maneuver.
How many individual positions have to be calculated for minor planets and satellites each day? My very rough guess is that this figure might approach one billion. It is that many times that Kepler's equation is being solved every day! Our modern computing machinery makes this easy. By contrast Kepler who used the best mathematical tools of the time - logarithmic tables - plus a human computer or two (who came under the name of mathematical assistant) would need a lot more time compiling his planetary tables.
But can you see the cultural decline of humanity here? Kepler set out to find a formula describing the majestic movements of planets across the sky and we end up using this very formula to keep track of all the trash we leave behind in space! I wonder where mankind will stand another 400 years further on...
One final remark (but only for those selected few who might not be able to sleep tonight if they didn't fully understand how the text in the Astronomia Nova correlates with the formula we use today). Instead of the area Kepler uses we have taken the corresponding angle on a circle and while Kepler took the zero position of the planets at the apocenter we now start counting from the pericenter. This is why Kepler talks about adding areas while we subtract angles.
This year has been designated the International Year of Astonomy as we celebrate the 400th anniversary of two seminal events: Galileo's telescopic observations of the sky and the publication of the Astronomia Nova by Johannes Kepler.
In his book Kepler describes in detail how he first found what we now call Kepler's Second Law (a line drawn from the Sun to the moving planet will sweep equal areas in equal times - actually a special case of the law of conservation of angular momentum) and toward the end of the book the discovery of Kepler's First Law (which states that the planets are orbiting the Sun on ellipses rather than circles). Kepler managed to achive this task by a combination of sound mathematical and astronomical knowledge, access to the most accurate observational data available at the time, hard work with a lot of trial and error but also by brilliant intuition.In reviews of popular science books you often read that the author managed to tell everything without using a single mathematical formula. Well, the Astronomia Nova should get high marks then. It is packed with mathematics but there is not a single formula to be found! How then does Kepler explain all the mathematics? By using lots of diagrams and even more text. As an example here is a line from chapter 47 (out of 70 chapters):
"You now have to determine the angle Bβδ in such a way that the product of its sine BC and half of αβ, which is to say the area of the triangle αBβ together with the sector Bβδ is equal to the area which before has been found from the time."
This sentence refers to a diagram (which I have greatly simplified - the original diagram contains many more lines and letters that are needed all throughout this chapter).Actually, the quote you just read and most likely didn't understand is nothing less than the first mentioning of what has since become known as the Kepler Equation. In modern textbooks it looks like this:
M = E - e sin(E)
Here you can see the beauty and power of modern mathematical notations. What has been a difficult to understand sentence in the Astonomia Nova is now condensed into a simple formula. So what does Kepler's equation tell us?
Let's start with M. This is called the Mean Anomaly and is an angle which increases linearly with time just like the hour hand of a clock would move once around in 60 minutes. If we know M we know the corresponding time and vice versa.
The e in front of the sine function is called eccentricity. It is 0 for a perfect circle and the larger it gets the more elongated the ellipse will be until after passing a value of 1 it is no longer a closed curve and turns into an open hyperbola. This is the trajectory of many comets and also of spacecraft leaving Earth on an interplanetary trajectory.
E is called the Eccentric Anomaly. This is an auxiliary angle which is directly related to yet another angle called the True Anomaly ν. It is this final angle which gives you the position of the planet along the ellipse. So knowing E is as good as knowing the position of the planet. If you know the eccentricity (which is one of the parameters of the orbit) and you have a position E then Kepler's equation will give you M and thus the corresponding time right away.
But usually you are interested in the opposite problem. You want to find the planet's position at a given time. It is now that our innocent looking little equation suddenly turns nasty. We know M and want to find E. E however appears not just by itself but also as an argument of the sine function. We cannot resolve Kepler's equation for E and get a straight answer. So what to do? Well you can always guess a value and see whether it fits and if it doesn't try another one. Mathematicians have found ways to do this systematically. Kepler knew of the Regula Falsi which you may or may not have heard about in school. But we have more efficient methods today. It typically takes just a few iterations to come up with a sufficiently precise answer to Kepler's equation.
In order to find a planet's position you need some information about it's orbit. We already learned of the eccentricity describing the shape of the ellipse. The size of the ellipse is given by the semi-major axis which in case of a circle would correspond to its radius. Three angles are needed to specify how the ellipse is oriented in space and finally we have to know where the planet was at a specific reference time. This set of data is usually referred to as the Kepler elements of an orbit.
Sophisticated methods exist today which generations of mathematicians have perfected to find the orbit elements from observed positions of the planet. How did Kepler do it? By having a clear three-dimensional understanding of the orbit in space he realized that certain privileged positions of the planets would help him find these orbit elements. He also had a good feeling about which observations to trust more than others. All this makes very interesting reading in the Astronomia Nova although most people will find this book slightly less easy to read than a paperback novel.
Once you know the Kepler elements of an orbit you can calculate the position of a planet via the Kepler equation for any time in the future or past. Well - not quite. More than one planet (and other objects) are orbiting the Sun and they all tug on each other via gravitation. There are also relativistic effects. This distorts our neat ellipses a little. For precise orbit calculations we now use methods of numerical integration.
The drawback of numerical methods is that you have to take one small step at a time and so it takes many time steps (and associated calculations) to finally arrive at the point in the future which you are interested in. By contrast you need just one calculation of the Kepler equation to find the position of the planet directly at any one time. Because this is a lot faster Keper's equation is usually employed for calculating the orbits of some tens of thousands of minor planets which zip around the solar system. Here Kepler elements are used which will give sufficiently accurate results for a few years after which they have to be updated.
Satellites orbiting the Earth are subject to a number of forces. The gravitational field of the Earth is not symmetric and has many bumps which influence the orbit. Satellites in low orbits are affected by atmospheric drag. Light pressure, Sun, Moon and relativity are also distorting our Kepler ellipses. Once again for high precision work you use numerical integration. But to do that you need accurate orbit information which you normally have for your own satellites but only rarely for others.
Most orbit information publicly available is in the form of so called Two-Line-Elements (TLE). This is essentially a set of Kepler elements with one extra number added to account for atmospheric drag. The procedure to calculate a satellite's position from TLEs runs over several pages and most of it is to account for the perturbing forces. But toward the end a few lines appear with a comment that here Kepler's equation is being solved.
Calculating satellite positions via TLEs is used by observers who want to know when to see the ISS or other satellites in the sky. But there are many other uses where the accuracy of this analytic method is adaquate. By far the biggest customers for TLEs (and thus for Kepler's equation) are the people who deal with space debris.
In addition to about 900 active satellites there are many other objects in orbit - inactive satellites, spent rocket stages, pieces left from explosions etc.
TLEs for more than 13000 objects are publicly available on the Internet (some 4000 more objects are individually tracked but are not publicly listed). In order to avoid a collision between an active satellite and some space debris you must check all the orbits a few days in advance and if there will be a close encounter perform an avoidance maneuver. How many individual positions have to be calculated for minor planets and satellites each day? My very rough guess is that this figure might approach one billion. It is that many times that Kepler's equation is being solved every day! Our modern computing machinery makes this easy. By contrast Kepler who used the best mathematical tools of the time - logarithmic tables - plus a human computer or two (who came under the name of mathematical assistant) would need a lot more time compiling his planetary tables.
But can you see the cultural decline of humanity here? Kepler set out to find a formula describing the majestic movements of planets across the sky and we end up using this very formula to keep track of all the trash we leave behind in space! I wonder where mankind will stand another 400 years further on...
One final remark (but only for those selected few who might not be able to sleep tonight if they didn't fully understand how the text in the Astronomia Nova correlates with the formula we use today). Instead of the area Kepler uses we have taken the corresponding angle on a circle and while Kepler took the zero position of the planets at the apocenter we now start counting from the pericenter. This is why Kepler talks about adding areas while we subtract angles.
