COMETS: VERY SPECIAL VISITORS Abstract

COMETS: VERY SPECIAL VISITORS Rosa M. Ros1, Sakari Ekko2 and Veselka Radeva3 EAAE Summer School Working Group (1Spain, 2Finland, 3Bulgaria)

Abstract A great comet is simply a beautiful spectacle. The late 20th century saw a lengthy gap without the appearance of any great comets, followed by the arrival of two in quick succession: Comet Hyakutake in 1996, followed by Hale-Bopp, which reached maximum brightness in 1997 having been discovered two years earlier. The first great comet of the 21st century was Comet McNaught which became visible to naked-eye observers in January 2007. It was the brightest in over 40 years. In the last few months of the year 2007 Comet Holmes had taken on a “special” appearance. This paper is based on a set of photographs that allow us to know something more about these fascinating bodies. This workshop about comets will also include simple models to help us understand them better. We will also carry out some simple calculations using measurements taken from the photos. The main objective of this workshop is to suggest a set of activities that each teacher can adapt for the students in order to promote comet observations. If the teachers are not experienced in photography, photos can be found on the internet. It is only necessary to know when the photos have been taken (day.month.year) in order to prepare similar activities to those we presented in this work.

INTRODUCTION Comets are the most attractive objects in the solar system because they are often very spectacular. Here are some examples of the presence of comets from some centuries ago (Figure 1 and 2). For instance, Halley’s Comet is a big bright comet which orbits around the Sun every 76 years on average. This is a periodical comet which follows an elliptic orbit. One very famous old recording of a comet is the appearance of Halley’s Comet on the Bayeux Tapestry (Figure 1), which records the Norman Conquest of England or in the Giotto’s well known Nativity (Figure 2). Halley’s Comet has an elliptic orbit. It is of note amongst solar system objects, because it is turning in the opposite direction to the planets, with 18º inclination to the ecliptic plane. It is a short period comet. Its next appearance will be in 2061.

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Figure 1. The Bayeux Tapestry which records the Norman conquest of England in AD 1066 show us the Halley’s Comet

Figure 2. Giotto di Bondone painting from about 1304 his Nativity and he drew a “Bethlehem Star” based on Halley’s Comet that he probably observed in 1301

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GENERAL CONCEPTS 1.- General ideas about comets The comets are small bodies in the solar system which orbit around the Sun. A comet is a solid body which sublimes in the surroundings of the Sun, then exhibits a visible coma (or atmosphere) and/or a tail. When the comet is far away from the Sun (5-10 au.) a coma appears around the nucleus. When the comet is closer the solar wind generates the tail which can be a dust tail and/or the ionic tail. Comets originate in the outer solar system, in the Oort cloud. The Oort cloud is thought to be a remnant of the original protoplanetary disc that formed around the Sun more than 4 billion years ago situated between 50,000 and 100,000 au. The most accepted hypothesis of its formation is that the Oort cloud’s objects initially formed much closer to the Sun as part of the same process that formed the planets, but that gravitational interaction with young gas giants such as Jupiter ejected them into extremely long elliptical or parabolic orbits. The current mass of the cloud (about 3 Earth masses) is only a small part of the mass of ejected material (50-100 Earth masses). Long-period comets are believed to originate in the Oort cloud. A lot of comets arrive close to the Sun following enlarged ecliptics orbits and they come back after thousands of years. When the orbits of these comets feel perturbations some of them can be lost in the solar system but others can suffer a reduction in the orbit size. In order to explain the existence of short-period comets, such as Halley’s Comet, Gerard Kuiper proposed the existence of Kuiper Belt. Short-period comets are thought to have originated in the Kuiper Belt, which lies beyond the orbit of Neptune. The Kuiper Belt is a flat frozen disc at 50 au. And the Oort cloud is a sphere of comets; their interior radius is about of 50,000 au. The short-period comets such as Halley’s Comet are formed in the Kuiper Belt (originally from the Oort cloud with a long–period orbit in the past). A long-period comet such as Hale-Bopp with a period of thousands of years has its origin in the Oort cloud. In the outer solar system, comets remain frozen and are extremely difficult to detect. When the comet approaches the inner solar system, solar radiation vaporises some materials of the nucleus. The streams of dust and gas thus released form a huge, extremely tenuous atmosphere around the comet called the coma, and the force exerted on the coma by the Sun’s radiation pressure and solar wind cause an enormous tail to form, which points away from the Sun. In 1986, Giotto photographed the nucleus of Halley’s Comet and observed the jets of evaporating material. The tail of dust is left behind in the comet’s orbit and normally forms a curved tail. The ion tail, made of gases, always points directly away from the Sun, as this gas is more strongly affected by the solar wind than the dust is. This ion tail follows magnetic field lines rather than an orbital trajectory. While the solid nucleus of a comet is generally less than 50 km in diameter, the coma may be larger than the Sun, and some ion tails may be larger than 1 au.

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The comets can be classified in a simple way in: periodical or non periodical. Short-period comets have orbital periods of less than 200 years. They usually orbit more or less in the ecliptic plane in the same direction as the planets. Their orbits typically take them out to the region of the outer planets, for example, Comet Halley’s is a little way beyond the orbit of Neptune. • Long-period comets have more elongated orbits than shorter period comets and their periods are about thousands or even millions of years. Their orbits take them far beyond the outer planets, and the plane of their orbits need not lie near the ecliptic. • Single-apparition comets are similar to long-period comets, but have parabolic or hyperbolic trajectories which will cause them to permanently exit the solar system after passing the Sun once. We do not consider any examples in the paper because they are not so common. •

Figure 3. The orbit of the comet of 1680, fit to a parabola, as shown in Isaac Newton’s Principia

2.- A little bit of ellipses The periodic comets return regularly. Their orbits are ellipses that obey the laws of Kepler. The comet orbits can be explained in terms of ellipses with high eccentricity. As the comets which are the subject of this work have elliptical orbits, we are going to limit this workshop to this kind of body. Comets have highly elliptical orbits. Here we introduce for consideration those essential elements of mathematics dealing with ellipses. An ellipse is a curve which generalizes the concept of circumference. If we have a dog with a rope tied to a pole, the dog traces a line when it is moving. If you hold the rope taut, we have a circumference of a circle: instead of the geometric place that joins all points of the plane, they are located at a distance of fixed r, called the radius, about a fixed point C, called centre (Figure 4). If we fix the dog to a rope through a ring which moves, and each end of this rope is fixed to two different posts, the dog runs in a curve called an ellipse (if the dog keeps the rope taut). That is to say, the ellipse is the geometric place that joins all points of the plane whose sum of distances is 2a, to two fixed points called focus F and F’

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Figure 4. Circumference of centre C and radius r

is constant. It is clear that if the two foci (or focuses) coincide with each other, the curve is now reduced to a circumference of a circle and the common point is the centre (Figure 5).

Figure 5. Ellipse with focus F and F’

Figure 6. Ellipse parameters

If we introduce reference axes in the ellipse (Figure 6) which we designated by a and b (semi-axes corresponding to the x and y axes). The distance from the origin to each of the foci is denoted by c. For any point the total distance to the two foci is 2a, in particular the distance from the focus F to the ellipse point (0,b), the intersection with the y axis is a. Therefore using the Pythagoras’ theorem in the triangle of Figure 6, it follows, _______ b =√ a2 – c2 We also define another parameter called eccentricity e, which allows us to assess whether the ellipse is very flat or not, _______ e = c/a = √1- (b/a)2 For the circumference of a circle e = 0 (because c = 0) and in general for ellipses 0 < c < a, and then 0 90º). Due to the effects of the projection, the angle on Table 1 is not the same that in planispheres (Figures 14 and 15). But it is clear that the angle which is the orbit with the ecliptic, in the case of Hale-Bopp’s comet is much smaller than the Hyakutake’s angle, according to the data of Table 1 (i = 88º.89