Calculating the Sun s Mass using Kepler s Laws of Planetary Motion. Anthony Enea. COSMOS UC Davis Cluster 9. July 28,2009

Transcription

1 Calculating Mass Using Orbital Computations 1 Calculating the Sun s Mass using Kepler s Laws of Planetary Motion Anthony Enea COSMOS UC Davis Cluster 9 July 28,2009

2 Calculating Mass Using Orbital Computations 2 Abstract This paper demonstrates a method of calculating the mass of any astronomical object using the objects that orbit them. We used data given to us by telescope exposures asteroid XP14, and used the data to substitute missing values in Kepler s three laws of planetary motion and Newton s law of gravitation to get the sun s mass. This paper also shows that lack of data leads to unfortunate outcomes, in which we checked with reliable sources.

3 Calculating Mass Using Orbital Computations 3 Calculating the Sun s Mass using Kepler s Laws of Planetary Motion If you are a curious soon-to-be astronomer and have always wondered how astronomers and physicists measure the masses of giant astronomical objects without using the scale you have in your bathroom, then you have found the right research paper. By reading this paper the method of calculating the mass of objects in space will become clear to you. In this paper we will describe to you how we calculated the mass of the Sun using the near-earth asteroid XP14, using 1) Kepler s three laws of planetary motion, which describe the shape and characteristics of a planet s orbit around the sun, and 2) Newton s laws of physics, which describe the motion and forces of objects in general. Using these conveniently easy to find laws, we can use orbital parameters such as the semi-major axis length and the orbital period to find the mass of the Sun, how can we trust these formulas to give us a correct answer? Johannes Kepler ( ), a German astronomer and mathematician, developed the three laws of planetary motion that are still used even today ( Johannes Kepler, 2009). Kepler s laws describe the characteristics of how one object orbits another. His first law states that all orbits are elliptical with a center of mass at one focus and each orbit has its own eccentricity (some ellipses are longer than others) (Weinstock, 1962). The Sun is a good example of where the focus is, since the location of the center of mass is related on the two objects masses, and the asteroid s mass compared to the sun s mass is tiny. Kepler s second law, the law of areas, states that the area the line between the two masses (sun and asteroid) sweeps out in a certain time is the same anywhere in its orbit (shown above), therefore explaining its acceleration as it gets closer to the sun. Kepler s third law, the law of periods, is that the period of an object squared is proportional

4 Calculating Mass Using Orbital Computations 4 to the semi-major axis of the ellipse cubed ( ). Later, Newton redefines this law by applying his law of gravitation to get a more precise equation ( ). Using software such as FindOrb, we can calculate the mass of the Sun. FindOrb measures a (semi-major axis length) and P (Period) from the observations we take (Nave, 2006a; Nave, 2006b). Now that you know Newton s more precise version of Keplers third law, where do we get the information needed to find the mass of the Sun? The first step is to take exposures of an asteroid, such as XP14, using a telescope. Then use software like CCDSoft, which shows and organizes exposures taken, and processes the images you took. Using the SKY program, which recognizes certain stars in the background of your image and gives you coordinates based on the stars position, to find the right ascension (R.A.) and declination (Dec.) of the asteroid in the exposures. Input those into the FindOrb program, which calculates the orbit using information given. FindOrb will find the semi-major axis and period of the orbit, which is all you need to find the mass of the sun (Whole procedure by Chris Taylor and Chris Fassnacht). So what is asteroid XP14? Wikipedia.com states, There were concerns that XP14 will possibly impact Earth later in the 21 st century. If this 2.5 trillion pound, 11 th magnitude, near- Earth asteroid with a 900 meter diameter hits the Earth, its impact would be the equivalent of 10,000 nuclear bombs exploding simultaneously, causing global devastation (Macey, 2006). July 3 rd 2006, the asteroid made its closest known approach yet, nearing the average distance the moon is away from Earth, and the exposures we took for this lab were taken on this date.

5 Calculating Mass Using Orbital Computations 5 Data The first data we received were 5 telescope exposures, taken on July 3 rd, 2006, when the asteroid was closest to the Earth. First, we dark subtracted the images in CCDSoft, taking all of the electronic noise and dead pixels out of the images. The asteroid appears as a line on the images we are using, taken by Paul Feldstein. Table of Paul Feldstein s Exposures

6 Calculating Mass Using Orbital Computations 6 Then, using the SKY program, which compares an objects coordinates with known coordinates of stars, we recorded the R.A and Dec. of the topmost part of each asteroid path. The top of each asteroid path will give us equal spaces between each exposure since each exposure has the same duration, therefore more organized data. The Epoch., R.A., and Dec. recorded were: R.A. Dec. Epoch. (Time taken) 1 st Exposure 00h 56m 32.09s +62º nd Exposure 00h 53m 57.63s +63º rd Exposure 00h 44m 54.74s +64º th Exposure 00h 39m 56.56s +64º th Exposure 00h 10m 24.36s +67º Table 1: Observations of asteroid position right ascension, declination and epoch (UTC) This data was input into FindOrb by copying in the R.A. and Dec. into the example. The period and semi-major axis of XP14 s orbit should be shown, but the information given by FindOrb was not shown because the Epoch. dates were too close together, adding to the number of possible elliptical orbits that could fit the given data (Problem made clear to me by Chris Taylor). Other possibilities could be bad data; one data point can mess up the entire calculation. If the recorded positions of the asteroid are too close together, then the amount of Figure 2: Orbit of Serius possible ellipses in which all those positions are located

7 Calculating Mass Using Orbital Computations 7 in will be higher, rather than when they are spread out. The ideal data point plot should be similar to that of figure 2. Unless we take exposures throughout the year, finding the exact orbit using FindOrb is very unlikely. Due to this unfortunate turn of events, we had to resort to other people s discoveries about the semi-major axis and period of XP14. The length of the aphelion is well-measured and is known to be AU, and the perihelion is , and the period is days ( 2004XP14 ). Now that we have the information we need, it is time to calculate the mass of the sun. Analysis Now that we have the data, we are ready to calculate the mass of the sun. First, we must add R1 and R2 together to get the major axis of the elliptical orbit. ( ). Major axis = 2.103AU. In order to get the semi-major axis, you need to divide the major axis by 2. (( ) or a) Now that we have the semi-major axis and the period we can now plug these into Newton s more accurate version of Kepler s third law. (( ) Zeilik, M.) The mass of the asteroid is so small compared to the mass of the sun; it can be taken out of the equation with no major effect. ( ) Next isolate the mass of the sun (M1). ( ) Next step is to use dimensional analysis to see what the end product will come out to be. Since, we will convert days into seconds, and AU into meters., this simplifies to M1=kg, so the end product will give us an answer in kilograms (With a big thanks to Augusta Abrahamse for demonstrating how to use dimensional analysis). Now it is time to

8 Calculating Mass Using Orbital Computations 8 convert the actual AU and days into meters and seconds days is equal to s, and AU is equal to. Substitute these numbers in to get: M1= This equation eventually works out to, and checked with sources such as Wikipedia and Google, this is close to the actual mass of the sun. The differences may be due to bad data or rounding, but this proves that Kepler knew what he was doing, and his equations work. Conclusion With this information, you can conclude that the mass of the sun s mass is around. Using this information, you can predict when our sun is going to go expand into a giant star. We can also conclude from this, that Kepler and Newton s equation was right, therefore the mass of any object can be found if the data is accurate. This can also support Newton s addition to Kepler s equation, relating the force of gravity between two objects and the intervals of time in which it orbits the ellipse.

9 Calculating Mass Using Orbital Computations 9 References 2004XP14 (n.d.) Retrieved July 14, 2009 from newton.dm.unipi.it web site: < Gravitation (n.d.) Retrieved July 12, 2009, from Wikipedia web site: < Johannes Kepler: His Life, His Laws and Times (2009) Retrieved July 15, 2009 from NASA web site: < Macey, R. (2006) Asteroid Set To Shave Earth. Retrieved July 25, 2009 from < Nave, C. (2006a) Circular Orbits. Retrieved July 12, 2009, from Georgia State University and Department of Physics and Astronomy, HyperPhysics database: < Nave, C. (2006b) Newton s Laws. Retrieved July 12, 2009, from Georgia State University and Department of Physics and Astronomy, HyperPhysics database: < Weinstock, R. (1962) Kepler s Third Law for Elliptical Orbits [Electronic Version]. American Journal of Physics, 30, Zeilik, M. (2002) Astonomy: The Evolving Universe 9 th Ed. Cambridge: Cambridge University

10 Press. Calculating Mass Using Orbital Computations 10