Astronomia nova
Astronomia nova is a book, published in 1609, that contains the results of the astronomer Johannes Kepler's ten-year-long investigation of the motion of Mars.
One of the most significant books in the history of astronomy, the Astronomia nova provided strong arguments for heliocentrism and contributed valuable insight into the movement of the planets. This included the first mention of the planets' elliptical paths and the change of their movement to the movement of free floating bodies as opposed to objects on rotating spheres. It is recognized as one of the most important works of the Scientific Revolution.
Background
Prior to Kepler, Nicolaus Copernicus proposed in 1543 that the Earth and other planets orbit the Sun. The Copernican model of the Solar System was regarded as a device to explain the observed positions of the planets rather than a physical description.Kepler sought for and proposed physical causes for planetary motion. His work is primarily based on the research of his mentor, Tycho Brahe. The two, though close in their work, had a tumultuous relationship. Regardless, in 1601 on his deathbed, Brahe asked Kepler to make sure that he did not "die in vain," and to continue the development of his model of the Solar System. Kepler would instead write the Astronomia nova, in which he rejects the Tychonic system, as well as the Ptolemaic system and the Copernican system. Some scholars have speculated that Kepler's dislike for Brahe may have had a hand in his rejection of the Tychonic system and formation of a new one.
By 1602, Kepler set to work on determining the orbit pattern of Mars, keeping David Fabricius informed of his progress. He suggested the possibility of an oval orbit to Fabricius by early 1604, though was not believed. Later in the year, Kepler wrote back with his discovery of Mars's elliptical orbit. The manuscript for Astronomia nova was completed by September 1607, and was in print by August 1609.
Structure and summary
In English, the full title of his work is the New Astronomy, Based upon Causes, or Celestial Physics, Treated by Means of Commentaries on the Motions of the Star Mars, from the Observations of Tycho Brahe, Gent. For over 650 pages, Kepler walks his readers, step by step, through his process of discovery. The work is divided into 5 parts, and contains a total of 70 chapters.Part 1
In the first part, Kepler examines the relationship between the various astronomical hypotheses that were in use at the time.In chapters 1-3, He shows that the heliocentric, geocentric and Tychonic models are all mathematically equivalent in that they predict the same angular positions of celestial object, and the variation in distance for any planet are also the same in all three models. This is so because the epicycle in the geocentric model plays the same role as the orbit of the Earth in heliocentric model and the orbit of the sun in Tycho's model. But the motion of the planets is observed to be non-uniform, even if we ignore the effects of the epicycle or the Earth's motion. Ptolemy and Copernicus had two different explanations for this. Ptolemy used an equant and eccentric circle, whereas Copernicus realized that he could combine two epicycles to explain the same effect. Kepler showed, however, that Copernicus' explanation is simply equivalent to an equant point with a non-circular orbit. The difference in the predictions between the two explanations were minor and are for practical purposes equivalent.
In chapters 4-6, Kepler considers a more physically plausible explanation for the non-uniform motion. He considers that the speed of the planet varies inversely with its distance from the sun. This explanation is shown, by calculation, to be consistent with the predictions of the equant or Copernicus' epicycles. But it requires Kepler to assume that the line of apsides for all planets intersect at the sun, whereas Ptolemy and Copernicus had assumed this point to be the center of the orbit of the earth/sun, which was referred to as the mean sun. The difference between the predictions is minor but is amplified if we also account for the effect of the Earth's orbit, in which case the difference could get as high as, which was certainly measurable.
Part 2
In part 2, Kepler introduces the vicarious Hypothesis, his first hypothesis to explain the motion of Mars.In chapters 7-10, Kepler tells the story of how he was introduced to the problem of Mars' orbit. Tycho and his assistant had been working on a theory of Mars, but they had failed to accurately account for the observed position of Mars. Tycho's observational data included 12 oppositions of Mars, for which he had determined its position in ecliptic longitude and latitude. They had managed to fit a theory to the observed ecliptic longitudes accurate to within 2 minutes of arc, yet it failed completely to account for the ecliptic latitudes. Kepler was then tasked with determining a more accurate theory to match this observational data.
His first step was to establish a precise definition of opposition. Since planets do not orbit in the same plane, in general they never reach precisely in angular separation. Ptolemy had assumed the planet reaches opposition when its ecliptic longitude is 180 degrees from the mean position of the sun. This definition ignores the ecliptic latitude, so when constructing the table of oppositions, Tycho's assistant suggested a correction to account for this, by instead measuring when the angle between the sun and one of the nodes along the ecliptic, was equal to the angle between planet and the opposite node measured along the path of the planet. But Kepler showed this correction to be erroneous for two reasons. First, the path of the planet as seen from the Earth is not the same as seen from the sun, and second, the ecliptic longitude of Mars as seen from the sun will not be the same as the ecliptic longitude of the Earth. The whole point of using oppositions is to eliminate the effect of the Earth's orbit, so that when we observe Mars from Earth, its position will be the same as if we observed it from the sun. So this error, which Kepler shows to be as high as 9 arcminutes, defeats the purpose of the correction.
The actual correction Kepler shows to be less than 1 minute of arc, which is smaller than the error in Tycho's observations, and for practical purposes can be ignored. So, opposition can be defined as the moment when the ecliptic longitude of the sun and Mars are apart. Although the ecliptic longitude of Mars is the same from the Earth and sun at this point, the same is not true for the ecliptic latitude. In the diagram above is the sun, is the Earth and is Mars. The line is in the plane of the ecliptic. The angle is the angle that Mars appears above the ecliptic when viewed from Earth; this is the ecliptic latitude. The angle is the ecliptic latitude viewed from the sun, which is smaller than. The relation between these angles tells us something about the ratio of the Earth-sun distance and the Mars-Sun distance.
In chapter 11, Kepler attempts to determine the parallax of Mars. As Mars is close to the Earth, its position in the sky will appear to change slightly as the observer's position changes throughout the day, even if it is otherwise stationary. This effect, called parallax, would be greatest when Mars is at opposition, since at that time Mars is at its closest point to the Earth. The existing estimates of the distance to the sun, based largely on Aristarchus' method, suggested the parallax could be as high as 6 minutes of arc, but Kepler's own attempts to determine parallax gave values that were less than 2 arc minutes.
In chapters 12-14, Kepler determines the longitude of Mars' ascending and descending nodes and the orbital inclination of Mars. To find the nodes, Kepler looks for observations of Mars where its ecliptic latitude is close to, then use interpolation to find the exact moment when it is zero and uses then use existing tables such as the Prutenic tables to compute the longitude of Mars at that time. Kepler located the ascending node at and the descending node at. These values are not precisely apart. The longitudes in the Prutenic tables were measured from the mean sun. Kepler argues that if they were measured from the sun's actual position instead the values would be apart.
The Prutenic tables also provided distances to the planets. This allows Kepler to solve the triangle in figure 1 above to compute heliocentric latitude from the observed geocentric latitude, from which he could deduce the orbital inclination of Mars by observing Mars when its latitude was greatest and computing its heliocentric latitude. He finds the orbital inclination to be. This also allowed Kepler to demonstrate the important fact that plane of Mars orbit does not wobble in any way as many theories before him had suggested. Using observations from various points, he shows the orbital inclination is constant.
In chapter 15, Kepler recomputes Tycho's 12 oppositions, so as to determine the precise moment Mars's ecliptic longitude is from the sun. For each observation, he determines the ecliptic longitude and latitude of Mars as seen from the Earth, and the time when opposition occurs.
In chapter 16, Kepler constructs his first model, the vicarious hypothesis, to account for the observations. This is a modification of the equant model of Ptolemy. In this model, the planet is assumed to move on a circular orbit, and the speed on the orbit varies in such a way that it appears uniform from some point called the equant. The line connecting the equant and the center of the circle is called the line of apsides, and it intersect the circle at two points, one where the planet moves at its fastest speed called the aphelion and the other where the planet moves at its slowest speed called the perihelion.
For his model, Ptolemy had assumed that the center of the circle lies exactly halfway between the equant and the point from which opposition is measured, this model is called bisected eccentricity. Kepler however considers the more general hypothesis that the center of the circle can be placed at any point along the line of apsides between the sun and the equant.
In the diagram above, let be the sun, be the center of the circular orbit, and be the equant point. The points and are the perihelion and aphelion respectively. Let be the position of Mars at a particular observation. The angle Kepler refers to as the true anomaly, and the angle the mean anomaly. For any observation, the true anomaly could be deduced if we knew the longitude of aphelion, by finding the difference between this and the longitude of the observation. The mean anomaly could be deduced if we knew the time when Mars it at aphelion, and by using the fact the Mars, viewed from the equant, traverses equal angles in equal times. If the true anomaly and the mean anomaly were known, we could likewise determine the location of the point by finding where the lines drawn from and intersect. For the purposes of calculation, we can take the length of the line to be.
Kepler's procedure is to take for 4 observations of Mars at opposition. By taking an initial guess for the longitude of aphelion and the time of aphelion, values could be computed for the mean anomaly and true anomaly of each observation, from which the location of Mars at each observation could be determined by the intersection of the lines and. If the 4 points do not lie on a circle, then the line of apsides is rotated about the point ; this shifts the values for the true anomalies, until all 4 points lie on a circle. Then if the center of the circle is not on the line of apsides, the line is rotated about the point until the point falls on the line of apsides; this shifts the values for the mean anomalies. But doing this also shifts the position of the points so that they no longer fall on a circle. This procedure is repeated again and again until all 4 points fall on a circle and the center of the circle falls on the line of apsides. This iterative process takes a long time to converge. In describing the procedure, Kepler writes:
If thou art bored with this wearisome method of calculation, take pity on me, who had to go through with at least seventy repetitions of it, at a very great loss of time.
At the end of the procedure, Kepler calculates the parameters for the model. He determines the longitude of aphelion as. The eccentricity of the circle is defined to be the distance from the center of the circle to the sun, divided by the radius of the circle, the value Kepler determines to be. The eccentricity of the equant is defined as divided by the radius of the circle, which he finds to be equal to. The sum of these values is referred to as the total eccentricity.
In chapter 17, Kepler makes a small correction for the fact that the longitude of aphelion and nodes are not constant but shift slowly over time.
In chapters 18-21, Kepler compares the theory to observations. First, he compares the longitude of the remaining 8 oppositions and finds that they all fit the predicted position of Mars to within Tycho's observational accuracy of two minutes of arc. This means that the vicarious hypothesis can be taken as an accurate theory for the true anomaly. Despite this remarkable accuracy, however, Kepler shows that the theory is false. He remarks:
Who would have thought it possible? This hypothesis, so closely in agreement with the observations, is nevertheless false.
Using the latitudes of the opposition and the latitude triangle from figure 1, Kepler is able to find the ratio of the Earth and Mars distances from the sun. The Earth-Sun distances are taken from an existing theory given by Tycho Brahe in his Progymnasmata, even though these values are not precisely correct, and the goal of the next part will be to determine a more accurate theory for the Earth's motion. The angles are determined by the latitude of the observations, and the angle is determined from the orbital inclination and the angle between Mars and the node. From this, the remaining sides can be determined, and the distances. By computing such distances, he obtained a lower and upper estimate for the eccentricity of Mars: -. The eccentricity found in the vicarious hypothesis is outside this range.
Kepler then examines another method for determining distances to Mars, by using observations of Mars when it is not at opposition and determining the longitude of Mars. The angle between the sun and Mars as viewed from Earth can be determined from observations. Tycho made many observations of Mars when it is not at opposition and determined the difference in ecliptic longitude between the sun and Mars in the sky. The angle between Mars and the Earth as viewed from the sun can be determined by calculating the heliocentric longitude of Mars from the vicarious hypothesis, and that of the Earth from Tycho's theory and taking the difference. And the distance from the Earth to the sun is given from Tycho's theory. Thus, the Earth, sun and Mars form a triangle, where two angles are known, and one side is given, the remaining sides and angles can be computed. In particular, we can determine the Earth-Mars distance. Computing these distances, Kepler once again finds an eccentricity closer to, half the value of the total eccentricity.
Kepler repeats the calculation where he substituted the mean sun in place of the true sun, to show that exactly the same thing arises in such case. So, the hypothesis of the true sun cannot be at fault. As a final recourse, Kepler considers what would happen if we substituted bisected eccentricity into the vicarious hypothesis, which is. When he compares this model to the observations of oppositions, he finds the error now increases to 8 minutes of arc, which is greater than Tycho's observational error. He writes:
Now, because they could not be disregarded, these eight minutes alone will lead us along a path to the reform of the whole of Astronomy, and they are the matter for a great part of this work.The inconsistency in determining the eccentricity means that at least one of the assumptions that went into constructing the vicarious hypothesis must be false: either the orbit is not circular, or there is no equant point a fixed distance away from the center of the circle.