Earth-moon-sun system: Phases and eclipses, Study notes of Reasoning

Download Earth-moon-sun system: Phases and eclipses and more Study notes Reasoning in PDF only on Docsity! NASE Publications EMS System: Phases and eclipses Earth-moon-sun system: Phases and eclipses Rosa M. Ros International Astronomical Union, Technical University of Catalonia (Barcelona, Spain) Summary The following work deals with moon phases, solar eclipses, and lunar eclipses. These eclipses are also used to find distances and diameters in the Earth-Moon-Sun system. Finally, the origin of tides is also explained. Goals - To understand why the moon has phases. - To understand the cause of lunar eclipses. - To understand why solar eclipses occur. - To determine distances and diameters of the Earth-Moon-Sun system. - To understand the origin of the tides. Relative positions The term “eclipse” is used for very different phenomena, but in all cases an eclipse takes place when one object crosses in front of another object; for this unit, the relative positions of the Earth and the Moon (opaque objects) cause the interruption of sunlight. A solar eclipse happens when the Sun is covered by the Moon when it is located between the Sun and our planet. This kind of eclipse always takes place during new Moon (figure 1). Lunar eclipses take place when the Moon crosses the shadow of the Earth. That is when the Moon is on the opposite side of the Sun, so lunar eclipses always occur at full moon phase (figure 1). The Earth and the Moon move along elliptical orbits that are not in the same plane. The orbit of the Moon has an inclination of 5 degrees with respect to the ecliptic (plane of Earth's orbit around the sun). Both planes intersect on a line called the Line of Nodes. The eclipses take place when the Moon is near the Line of Nodes. If both planes coincided, the eclipses would be much more frequent than the zero to three times per year. NASE Publications EMS System: Phases and eclipses Fig.1: Solar eclipses take place when the Moon is located between the Sun and the Earth (new Moon). Lunar eclipses occur when the Moon crosses the shadow cone of the Earth (that is, the Earth is located between the Sun and the full Moon). Masks models Model of Hidden Face The Moon has two movements: rotation and translation which has approximately the same duration, that is to say about four weeks. This is the reason that from the Earths we can see always the same half lunar superfice. We will see this situation with a simple model. We begin by placing the volunteer who plays the role of Earth and only one "Moon" volunteer with a white mask. We place the "Moon" volunteer in front of Earth, looking to the Earth, before starting to move. So if the Moon moves 90 degrees in its orbit around the Earth, it also must turn 90 degrees on itself and therefore will continue looking in front of the Earth, and so on. We will ask to the Earth volunteer if he/she can see the same face of the Moon or can see a differnet part. We repeat the same situation four times, always moving 90º. It is evident that each 90º, that is to say each week, the Earth can see always the same part of the moon, the back of the head of the voluteer is never visible. Moon Phases model To explain the phases of the Moon it is best to use a model with a flashlight or with a projector (which will represent the Sun) and a minimum of five volunteers. One of them will be located in the center representing the Earth and the others will situate themselves around "the Earth" at equal distances to simulate different phases of the Moon. To make it more NASE Publications EMS System: Phases and eclipses Fig.5a and 5b: Lunar eclipse simulation. Fig. 6: Photographic composition of a lunar eclipse. Our satellite crosses the shadow cone produced by the Earth. Reproducing the eclipses of the Sun The model is placed so that the ball of the Moon faces the Sun (it is better to use the projector or the flashlight) and the shadow of the Moon has to be projected on the small Earth ball. By doing this, a solar eclipse will be reproduced and a small spot will appear over a region of the Earth (figures 7a, 7b and 8). NASE Publications EMS System: Phases and eclipses Fig. 7a and 7b Solar eclipse simulation It is not easy to produce this situation because the inclination of the model has to be finely adjusted (that is the reason why there are fewer solar than lunar eclipses). Fig.8: Detail of the previous figure 7a. Fig. 9: Photograph taken from the ISS of the solar eclipse in 1999 over a region of the Earth’s surface. NASE Publications EMS System: Phases and eclipses Observations  A lunar eclipse can only take place when it is full Moon and a solar eclipse when it is new Moon.  A solar eclipse can only be seen on a small region of the Earth’s surface.  It is rare that the Earth and the Moon are aligned precisely enough to produce an eclipse, and so it does not occur every new or full Moon. Model Sun-Moon In order to visualize the Sun-Earth-Moon system with special emphasis on distances, we will consider a new model taking into account the terrestrial point of view of the Sun and the Moon. In this case we will invite the students to draw and paint a big Sun of 220 cm diameter (more than 2 meters diameter) on a sheet and we will show them that they can cover this with a small Moon of 0.6 cm diameter (less than 1 cm diameter). It is helpful to substitute the Moon ball for a hole in a wooden board in order to be sure about the position of the Moon and the observer. In this model, the Sun will be fixed 235 meters away from the Moon and the observer will be at 60 cm from the Moon. The students feel very surprised that they can cover the big Sun with this small Moon. This relationship of 400 times the sizes and distances is not easy to imagine so it is good to show them with an example in order to understand the scale of distances and the real sizes in the universe. All these exercises and activities help them (and maybe us) to understand the spatial relationships between celestial bodies during a solar eclipse. This method is much better than reading a series of numbers in a book. Earth Diameter 12 800 km 2.1 cm Moon Diameter 3 500 km 0.6 cm Distance Earth-Moon 384 000 km 60 cm Sun Diameter 1400 000 km 220 cm Distance Earth-Sun 150 000 000 km 235 m Table 2: Distances and diameters of system Earth-Moon-Sun Fig. 10: Sun model. Fig. 11: Observing the Sun and the Moon in the model. NASE Publications EMS System: Phases and eclipses Nowadays we know that he was slightly wrong, possibly because it was very difficult to determine the precise timing of the quarter moon. In fact this angle is  = 89 º 51 ', but the process used by Aristarchus is perfectly correct. In figure 15, if we use the definition of secant, we can deduce that cos  = ES/EM where ES is the distance from the Earth to the Sun, and EM is the distance from the Earth to the moon. Then approximately, ES = 400 EM (although Aristarchus deduced ES = 19 EM). Relationship between the radius of the Moon and the Sun The relationship between the diameter of the Moon and the Sun should be similar to the formula previously obtained, because from the Earth we observe both diameters as 0.5 º. So both ratios verify RS = 400 RM Relationship between the distance from the Earth to the Moon and the lunar radius or between the distance from the Earth to the Sun and the solar radius Aristarchus supposes the orbit of the moon as a circle around the Earth. Since the observed diameter of the Moon is 0.5 degrees, the circular path (360°) of the Moon around the Earth would be 720 times the diameter. The length of this path is 2 times the Earth-Moon distance, i.e. 2 RM 720 = 2  EM. Solving, we find EM = (720 RM)/ Using similar reasoning, we find ES = (720 RS)/ This relationship is between the distances to the Earth, the lunar radius, the solar radius and the terrestrial radius. Relationship between the distances from the Earth to the Sun and to the Moon, the lunar radius, the solar radius and the terrestrial radius. During a lunar eclipse, Aristarchus observed that the time required for the Moon to cross the Earth's shadow cone was twice the time required for the Moon's surface to be covered (figures 16a and 16b). Therefore, he concluded that the shadow of the Earth's diameter was twice the diameter of the Moon, that is, the ratio of both diameters or radius was 2:1. Today, it is known that this value is 2.6:1. NASE Publications EMS System: Phases and eclipses Fig. 16a: Measuring the cone of shadow. Fig.16b: Measuring the diameter of the Moon. Final Summary Taken into accon the last results, Then (figure 17) Fig. 17: Shadow cone and relative positions of the Earth-Moon-Sun system we deduce the following relationship: x /(2.6 RM) = (x+EM) / RE = (x+EM+ES) / RS where x is an extra variable. Introducing into this expresion the relationships ES = 400 EM and RS = 400 RM, we can delete x and after simplifying we obtain, RM = (401/1440) RE This allows us to express all the sizes mentioned previously as a function of the Earth’s radius, so RS= (2005 /18) RE, ES = (80200 / RE, EM = (401 /(2 RE where we only have to substitute the radius of our planet to obtain all the distances and radii of the Earth-Moon-Sun system. Measurements with students It's a good idea to repeat the measurements made by Aristarchus with students. In particular, we first have to calculate the angle between the Sun and the quarter moon. To make this measurement it is only necessary to have a theodolite and know the exact timing of the quarter moon. NASE Publications EMS System: Phases and eclipses So we will try to verify if this angle measures = 87º or = 89º 51’ (although this precision is very difficult to obtain). Secondly, during a lunar eclipse, using a stopwatch, it is possible to calculate the relationship between the following times: "the first and last contact of the Moon with the Earth's shadow cone", i.e., measure the diameter of the Earth’s shadow cone (figure 17a) and "the time necessary to cover the lunar surface," that is a measure of the diameter of the moon (figure 20b). Finally, it is possible to verify if the ratio between both is 2:1 or is 2.6:1 or it is different. The most important objective of this activity is not the result obtained for each radius or distance. The most important thing is to point out to students that if they use their knowledge and intelligence, they can get interesting results with few resources. In this case, the ingenuity of Aristarchus was very important to get some idea about the size of the Earth-Moon-Sun system. It is also a good idea to measure with the students the radius of the Earth following the process used by Eratosthenes. Although the experiment of Eratosthenes is well known, we present here a short version of it in order to complete the previous experience. Eratosthenes’ experiment, again Eratosthenes was the director of the Alexandrian Library. In one of the texts of the library, he read that in the city of Syena (now Aswan) the day of the summer solstice, the solar noon, the Sun was reflected in the bottom of a well, or what it is the same the stick did not produce shadow. He noted that the same day, at the same time, a stick produced no shadow in Alexandria. From this, he deduced that the surface of the Earth could not be flat, but it should be a sphere (figures 18a and 18b) Fig. 18a an 18b: In the flat surface the two sticks produce the same shadow, but when the surface is corved sahdows are differetn. Consider two stakes placed perpendicular to the ground, in two cities on the Earth’s surface on the same meridian. The sticks should be pointing toward the center of the Earth. It is