Determining solar characteristics using planetary data

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1 Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation I will be examining the Sun and its physical chaacteistics. I will use a database poided by the Euopean Space Agency, found at to acquie aious pieces of infomation about the planets in the Sola System and then use that to detemine some of the Sun s chaacteistics though the laws of astophysics. Featue 1: Mass The most basic chaacteistic about the Sun that I can find is its mass. The mass of the Sun must fist be found befoe it is possible to detemine chaacteistics based on gaitational foce. I plan to detemine the mass of the Sun using data fom the planets. This can be done though the application of Johannes Keple s laws of obital motion. Keple s thid law states that: The squae of a planet s obital peiod is diectly popotional to the cube of its semi majo axis. This law can be demonstated using the planets in ou Sola System and thei obits aound the Sun. I found a gaph on the intenet which shows the accuacy of Keple s thid law in ou Sola System. Page 1 of 8 Physics teache suppot mateial 1

2 The obital peiod of a body is the time it takes to complete one full obit. The semi majo axis is half of the longe diamete of an obit s elliptical path. Any popotionality statement can be expessed in an equation though the addition of a constant. In this case, the constant must elate to the gaitational foce which guides the obits and thei cicula paths. The full equation fo Keple s thid law is gien as: P 4 a GM ( M) 3 1 Whee P is the obital peiod, G is the uniesal gaitational constant (6.67x10 11 m 3 kg 1 s ), M 1 and M ae the masses of the two celestial bodies and a is the semi majo axis of the obit. This equation can then be eaanged to find the sola mass, M1: Page of 8 Physics teache suppot mateial

3 M 1 4 a 3 P G M Now, using data fom the planetay database, I can calculate the mass of the Sun. To impoe the accuacy of the esult, I did this calculation fo each of the eight planets. Sample calculation using data fom Mecuy: M M ( ) ( ) ( ) kg 3 These ae the esults I obtained fo the sola mass using data fom each of the planets: Planet Sola Mass (kg) Mecuy x10 30 Venus x10 30 Eath x10 30 Mas x10 30 Jupite x10 30 Satun x10 30 Uanus x10 30 Neptune x10 30 All the esults ae ey simila when ounded. The main diffeence occued when going fom the smalle and lighte inne planets to the lage and heaie oute ones. The geate mass of these planets changed the alue calculated slightly. I decided to take an aeage mass fom these esults and got my final alue fo the sola mass as x10 30 kg. Page 3 of 8 Physics teache suppot mateial 3

4 Featue : Suface Gaity Afte successfully detemining the mass of the Sun though Kepleian calculations, the Sun s gaitational foce can be calculated. The suface gaity of a body is the acceleation due to the foce of its gaity when at its suface. The mathematical fomula fo acceleation due to gaity is deied fom Newton s law of uniesal gaitation. Newton s law is shown in the pictue below. Then, taking Newton s second law, F get: mg and setting the two foces equal to each othe we mg GMm The masses then cancel out allowing one to sole fo the acceleation due to gaity using the following fomula: g GM Whee g is the suface gaity, G is the gaitational constant, M is the mass of the body and the distance fom the body s cente of gaity. To calculate the suface gaity, the alue fo must be set equal to the body s adius. In the last section I detemined the mass of the Sun. Its Page 4 of 8 Physics teache suppot mateial 4

5 adius can be found though measuing its angula size in the sky and then, knowing the distance fom the Eath to the Sun, the adius can be calculated tigonometically. I was unable to do this myself, so I obtained the adius of the Sun fom my database, as km. Using this data, the suface gaity of the Sun can be calculated. g g ( )( ) 74.3ms ( ) As a compaison, the suface gaity on Eath is 9.81ms, almost 8 times less. Its ey lage gaitational foce is how the Sun manages to keep all the planets in obit. Featue 3: Escape Velocity The escape elocity of a celestial body is the elocity equied by an unpoweed object to escape that body s gaitational field. At escape elocity, the gaitational potential enegy of the object and its kinetic enegy will equal zeo when added. The fomula fo escape elocity can be deied though this. Kinetic enegy has the fomula E Ek g 1 m and gaitational potential enegy is gien by GM M 1. When these two ae added, they must equal zeo, so: 1 m GM M 1 0 Page 5 of 8 Physics teache suppot mateial 5

6 This now allows us to eaange this fomula to sole fo, the escape elocity. GM Now the alues that we obtained can be inseted into the equation ( )( ) ms This alue I obtained is the elocity an object fied fom the suface of the Sun to be fied in ode to escape its gaitational field about 617.7kms 1. Howee, this alue does not hae many pactical applications as we do not fie objects fom the suface of the Sun. A much moe useful alue to calculate would be the elocity equied to escape sola obit when launched fom the Eath. This allows us to launch pobes into the intestella medium. To calculate this alue, the Eath s aeage distance fom the Sun, one astonomical unit, must be set as the alue in the equation ( )( ) 4118ms In ode to escape the Sun s gaity when launched fom the Eath, a elocity of 4kms 1 is equied. This alue can be used if pobes ee need to be sent on a sola escape tajectoy. Page 6 of 8 Physics teache suppot mateial 6

7 Conclusion Using data poided in an online database, I was able to detemine the mass, gaitational foce and escape elocity of the Sun. To check the accuacy of the esults I calculated, I compaed them to the actual accepted alues. The mass of the Sun that I obtained was x10 30 kg. The actual accepted alue is x10 30 kg. My esult was only 0.015% highe than the accepted alue. This shows that though the application of Keple s thid law, celestial masses can be calculated with a ey good amount of accuacy. The next chaacteistic that I calculated was the suface gaity of the Sun. The alue I obtained was 74.3ms 1. The accepted alue is 74.0ms 1. My alue is 0.15% highe which is once again ey close. The eason that my esult is highe is because I used the mass of the Sun that I calculated in the fist pat, which was also slightly highe than the actual mass. Howee, my calculations wee still ey accuate. The final chaacteistics I calculated wee the escape elocities fom the suface of the Sun and also the sola escape elocity fom Eath. Fo the escape elocity fom the suface of the Sun, I got a alue of 617.7kms 1. This is identical to the accepted alue, which means that I was able to pefom my escape elocity calculation ey accuately. The esult that I calculated fo the sola escape elocity fom Eath was 4.1kms 1. This is also identical to the accepted alue. My escape elocity calculations wee extemely accuate. Oeall, the alues I obtained though my calculations in this inestigation wee ey accuate with egads to the actual alues and I am pleased with how it tuned out. I was able to apply Keple s and Newton s laws of astophysics to the planets and the Sun in ou Sola System well and though this I was able to confim the accuacy of these laws. Page 7 of 8 Physics teache suppot mateial 7

8 Ealuation I chose to do this inestigation because of a pesonal inteest in the field of astophysics. I chose to do ou Sola System because it is the system that we ae most familia with and we hae an abundance of data to do with the celestial bodies in it. The Sun, being the focal point of the Sola System, was the best choice fo me to conduct my inestigation on. I knew that I could use the obital data of the planets to calculate aious chaacteistics of the Sun. Fo my database, I chose the one poided by the Euopean Space Agency because it has ey accuate and pecise infomation with easy accessibility. The database poed ey useful thoughout my inestigation and gae me ey accuate esults in the end. Thoughout my inestigation, I had to apply aious physical laws. In ode to conduct the fist pat of my inestigation I had to familiaize myself with Keple s thid law, a topic that we only studied biefly in class and I did not know too much of. I was able to use the law popely and accuately to calculate my data. Newton s laws of gaitation wee ones that we had studied moe extensiely so I was also able to apply them in a pope manne and effectiely. The accuacy of my esults shows a good application of these laws. I am pleased with how the inestigation tuned out and the esults I managed to obtain. If I could add moe to this inestigation I would pobably look into doing futhe calculations of diffeent factos, examining chaacteistics such as the Sun s suface tempeatue, luminosity and enegy output. I would also pobably do moe calculations fo my cuent factos, such as expanding my sola escape elocities to include those fom all the planets. In this inestigation I only looked into a small faction of the sola chaacteistics aailable. Howee, I think that what I did with my inestigation went ey well. Page 8 of 8 Physics teache suppot mateial 8

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