Solving Problems of Advance of Mercury s Perihelion and Deflection of. Photon Around the Sun with New Newton s Formula of Gravity

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1 Solving Poblems of Advance of Mecuy s Peihelion and Deflection of Photon Aound the Sun with New Newton s Fomula of Gavity Fu Yuhua (CNOOC Reseach Institute, fuyh945@sina.com) Abstact: Accoding to the new Newton's fomula of gavity (the oiginal law of gavity plus a coection tem), i.e., impoved Newton's fomula of gavity, applying the methods of classical mechanics to solve the poblem of advance of Mecuy's peihelion and the poblem of deflection of photon aound the Sun espectively, and the esults ae the same as given by geneal elativity. Pointing out that the futhe topic is based on deiving oiginal law of gavity and oiginal Newton s second law with law of consevation of enegy to deive the new Newton's fomula of gavity (the impoved Newton's fomula of gavity) with law of consevation of enegy. To ealize the pupose that patially eplacing elativity and solving some poblems that cannot be solved by elativity with the methods of classical mechanics. Key wods: Law of gavity, new Newton's fomula of gavity (impoved Newton's fomula of gavity), advance of Mecuy's peihelion, deflection of photon aound the Sun, law of consevation of enegy Intoduction In efeences [, ], the new Newton s fomula of gavity (the oiginal law of gavity plus a coection tem), i.e., the impoved Newton's fomula of gavity, was pesented; but it did not pesent the detailed pocess to solve the poblem of advance of Mecuy's peihelion and the poblem of deflection of photon aound the Sun espectively with the methods of classical mechanics. While, in this pape the detailed pocess will be given (the esults ae the same as given by geneal elativity). The impoved Newton's fomula of gavity is as follows GMm F 3G M c 4 mp () whee: G is gavitational constant, M and m ae the masses of the two objects, is the distance between the two objects, c is the speed of light, p is the half nomal chod fo the object with mass m moving aound the object with mass M along a cuve, and the value of p is given by: p a (-e ) (fo ellipse), p a (e -) (fo hypebola), p = y / (fo paabola). Solving the poblem of advance of Mecuy's peihelion with new Newton s fomula of gavity In classical mechanics, acted by the cental foce, the obit diffeential equation (Binet s fomula) eads h u whee: u. ( u" u) F m ()

2 As deiving Eq.(), it aleady gives h GMp (3) Substituting Eq.() and Eq.(3) into Eq.(), we have the following equation of planet s movement aound the Sun 3GMu u" u (4) p c Fo ellipse, p a (e -), thus the appoimate solution fo Eq.(4) is as follows GM cos( 3GM u [ e ) ] (5) a( e ) c a( e ) c Hence, the value of fo advance of planetay peihelion fo one cicuit is as follows 3 4 a (6) T c ( e ) whee: T, a, and e ae obital peiod, semi-majo ais and eccenticity espectively. Obviously, this esult is the same as given by geneal elativity. In addition, accoding to Eq.(), fo poblem of planetay motion aound the Sun, the impoved Newton's fomula of gavity eads GMm 3G F M ma( e 4 c ) Solving the poblem of deflection of photon aound the Sun with new Newton s fomula of gavity As solving this poblem by using the impoved fomula of gavity, the method to be used is the same as pesented in efeences [3], in which the oiginal law of gavity was used. Fig. Deflection of photon aound the Sun Supposing that m epesents the mass of photon, fo the eason that it will be eliminated, so it is not necessay to give its value. As shown in Fig., epesents the neaest distance between the photon and the cente of the Sun, fo the eason that the deflection is vey small, so actually

3 the value of is the same as the photon is not deflected. When the photon is located at (,y) (the value of y is measued fom point P in Fig.), the foce acted on photon is as follows F whee: F F (7) / ( y ) GMm 3G M y c ( y ) mp Because mv F dt F dy v y c F dy Theefoe v GM dy 6G M p dy c y c y 3/ 3 5/ ( ) ( ) Afte calculating, it gives v GM 4G M p (8) c c 3 3 Hence, the deflection angle is as follows v GM 4G M p tg (9) c c c 4 3 While, the value of should be detemined by iteation method. Befoe detemining the value of, fistly we will validate that the value of the second tem in Eq.(9) is equal to the value of the fist tem, that means that the deflection given by the second tem in Eq.(9) is equal to that given by the fist tem. As solving poblem of deflection of photon aound the Sun with geneal elativity, the photon s obit is a hypebola, and its equation is as follows GMu (sin) u ucos () c whee: u Hence, the ecipocal of the half nomal chod p is as follows p u / GM c ()

4 Substituting this value of the half nomal chod p into Eq.(9), it gives 4GM 4GM () c c R s whee: R s is the adius of the Sun. Thus, the value of the second tem in Eq.(9) is eally equal to the value of the fist tem. Now we detemine the value of in Eq.(9) by iteation method. Suppose KGM (3) c In ode to apply iteation method, the elationship between and p should be given. Consideing two staight lines, the fist one is, the second one is passing though the oigin O and the fist quadant, and it makes an angle of / with the positive diection of Y ais; the value of the half nomal chod p is equal to the distance between the oigin O and the intesection of the two staight lines. Supposing that the intesection of the two staight lines is the point P (it is not shown in Fig.), then its coodinates ae (, p ). Fom the tiangle fomed by the thee points of oigin O, P, and P, it gives sin p p (4) Now, consideing the esult of the value of given by the oiginal law of gavity, it gives K Hee, the deflection angle is as follows GM c Fom Eq.(4), its coesponding half nomal chod p is as follows c p GM

5 Substituting the value of p into Eq.(9), it gives 6GM c Namely K 6 Similaly, the values of K, K 3, and the like ae as follows: , 4.4, 3.88, 4.95, , 4.34, , 4.59, 3.997, 4.5, , 4.4, , 4., 4., 4.; finally it gives K 4 This esult is also the same as given by geneal elativity. Accoding to Eq.(), fo poblem of deflection of photon aound the Sun, the impoved Newton's fomula of gavity eads GMm.5GMm F 4 whee: is the shotest distance between the photon and the Sun, if the light and the Sun is tangent, it is equal to the adius of the Sun. The inteesting fact is that, fo this poblem, the maimum gavitational foce given by the impoved Newton s fomula of gavity is.5 times of that given by the oiginal Newton s law of gavity. 3 Futhe topic In efeences [4-6], the oiginal law of gavity and the oiginal Newton s second law have been deived with law of consevation of enegy, based on this, the futhe topic is to deive the new Newton's fomula of gavity (the impoved Newton's fomula of gavity), i.e., Eq.(), with law of consevation of enegy. And, finally, to ealize the pupose that patially eplacing elativity and solving some poblems that cannot be solved by elativity with the methods of classical mechanics. Refeences Fu Yuhua, Impoved Newton s fomula of univesal gavitation, Zianzazhi (Natue Jounal), (), Floentin Smaandache, V. Chistianto, Fu Yuhua, R. Khapko, J. Hutchison. Unfolding the Labyinth: Open Poblems in Physics, Mathematics, Astophysics, and Othe Aeas of Science. Heis-Phoeni, 6, Kittel C., Knight W. D. and Rudeman M. A., Mechanics. Bekeley Physics Couse Vol., McGaw-Hill, Fu Yuhua, Deiving Impoved Newton s Second Law and the Law of Gavity at One Time with Fom of Factal Fomula, Engineeing Science. 3,Vol.5,No.6,55-58

6 5 Fu Yuhua, Epanding Newton Mechanics with Neutosophy and Quad-stage Method New Newton Mechanics Taking Law of Consevation of Enegy as Unique Souce Law, Neutosophic Sets and Systems, Vol.3, 4 6 Fu Yuhua, New Newton Mechanics Taking Law of Consevation of Enegy as Unique Souce Law, Science Jounal of Physics, Volume 5, Aticle ID sjp-3, Pages, 5, doi:.737/sjp/3