A New Theory for Celestial Orbits Necat Taşdelen Dipl.Mech.Engineer, Istanbul-Turkey ABSTRACT Since Kepler s time (1609) we are educated to believe: Orbits of the planets are elliptical, with the Sun at one focus of this trajectory; that the ray Planet-Sun sweeps out equal area in equal interval of time; that a period law is valid. I claim the contrary. Clearly, it is difficult to agree with novelties, when the community is trained on wrong direction since 400 years. But, mathematics can clarify our understanding. According Newton s laws the orbits of celestial bodies are spiraled. After presenting the mathematical foundations I argue that our educational system about Keplerian math should be corrected. Keywords: Kepler laws, elliptic orbits, Newton laws, spiraled orbits, Celestial orbits 1. INTRODUCTION Fig.1 explains the vector components in physics Fig 1: Velocities, forces, works are all directional values: vectors. 2. CONSIDER NEWTON S (F*dt=m*dv) expression; (1) dwr=dfr*lr+fr*dlr dwp=dfp*lp+fp*dlp and we write Where F is a vector and its components are Fradial=Fr and Fperpendicular=Fp and, Work=F*Lwhere (L) is the displacement length;when differentiated (2) dwp=d(m*dvp/dt)*lp+(m*dvp/dt)*dlp In physics the work in the perpendicular direction to the attraction field equal zero. Fig.2 241
Fig 2: No work is done in the direction perpendicular to the attraction field direction So, dwp=0 and for this,we have to write dvp/dt=0 (3) 1/2*m*Vt1^2+m*a1*r1+m*Ct1=1/2*m*Vt2^2+m*a2*r 2+m*Ct2=Ct0 (7) (Vt) is a vector and Vt^2=Vradial^2+Vperpendicular^2. which means, when integrating That is: Vp=Ct (4) Kepler s law about the equality of swept out area in equal interval of time is not correct. Kepler says: (r* Vp =Ct) (area law which is pronounced as r and Vp variables) Newton laws say: (Vp = Ct) (no area law) What a claim! This is the basic of the following lines. 3. ON OTHER HAND, (TRAJECTORIES SHAPE) We know: Energy total= E.kinetic+E.potential =Constant (conserved) and we write 1/2*m*Vt^2+m*a*r+1/2*I*w^2= m*r*dvr/dt=ct0 (5) Also, knowing 1/2*I1*w1^2 =m*ct1=1/2*i2*w2^2=m*ct2 (6) as innate (no intervention since the existence) We write, Eliminating m*ct1=m*ct2 on both sides of the equality We write for any position of the planet in its solar system 1/2*m*Vr1^2+1/2*m*Vp1^2+m*a1*r1=1/2*m*Vr2^2+ 1/2*m*Vp2^2+m*a2*r2 (8) As from (4) 1/2*Vp1^2=1/2*Vp2^2 =Ct3, the expression (5) is written as (1/2*m*Vr^2+ m*ct3)+ m*a*r + m*ct1=m*r*dvr/dt=ct0 (9) Dividing both sides by (1/2*m), we have consecutively Vr^2+2*Ct3+2*a*r+2*Ct1=2*r*dVr/dt (Vt^2-Vp^2)+2*Ct3+2*a*r+2*Ct1=2*r*dVr/dt Vr^2+2*a*r+2*Ct1=2*r*dVr/dt (10) The differential form of (10) is r ^2+2*a*r+K=2*r*r (11) where (K=2*Ct1= I*w^2/m). The solution of (11) is r=-a*t^2+a*t*t+z(t) (12) 242
where T=Total life-time of the celestial body and the final equations are: Distances equation r=-a*t^2+a*t*t+z(t) (13) Radial velocity =Vr r =dr/dt=-2*a*t+a*t+z (14) Radial acceleration=dvr/dt) r =d(dr/dt)/dt=-2*a +Z (15) Perpendicular Velocity Vp=Constant (4) Perpendicular acceleration dvp/dt=0 (3) Equation (13) does not show any sign of ellipse, but a parabola on Cartesian, a spiraled shape projection on Polar. And that means the shapes of the orbits are not elliptical. The equation (12) is the equation of the planetary motion mathematically. Kepler says: planetary orbits are elliptical Newton laws say: planetary orbits are spiraled; Big difference 4. WHAT IS Z (T)? Consider the equation (11) r ^2+2*a*r+K=2*r*r where (K=2*Ct1= I*w^2/m). r=-a*t^2+a*t*t+z where, when t=0, r=zo constant The solution of (11) is (12). When displayed, and simplified a^2*t^2-2*a^2*t^2+2*a^2*t*t+2*a*z+k=-4*a*z written When (t=0 ) we write also: a^2*t^2+2*a*z+k=-4*a*zo K+a^2*T^2=-6*a*Zo Zo=-(K+a^2*T^2)/(6*a) replacing (Zo) in (13) r=-a*t^2+a*t*t-(k+a^2*t^2)/(6*a) a^2*t^2-2*a*(k+a^2*t^2)/(6*a)=4*a*(k+a^2*t^2)/(6*a) 3*a^2*T^2-6*a(..)/(6*a)=2*6*a(..)/(6*a) 3*a^2*T^2=3*( ) a^2*t^2=k+a^2*t^2 then, K=0 which means: when t=0, w=0, (K=I*w^2/m=0) Zo=-a*T^2/6 is found. We write the final distance equation (13) as: r=-a*t^2+a*t*t-a*t^2/6 (16) Equation (16) shows a multilayer spirals with variable amplitude. When (t=0) the distance is negative, indicating that the planets are born from the Sun: a maturation time inside the Sun, then (r=0) within a special time, then (w) is obtained. Orbiting life-time starts with the maturation. (Kind of pregnancy) Fig.2 shows Earth orbit on Cartesian (left) and its projection on polar (right),shortened to T=3 cycles. is Fig 3: Earth real orbit: birth to death; cycles reduced to 3 cycles for easy understanding of spiral shape. 243
5. COMMENTING THE MATHEMATICAL RESULTS FOR PHYSICS - Planets do not orbit the Sun on an elliptical trajectory. - The Sun is not at a focus of such elliptic orbit, but at the barycenter of the spirals.(heliocentric) - There is only one extreme (r) on these orbits; no aphelion, no perihelion. They do not exist. - As the Sun is travelling on its trajectory around the Milky Way, the spirals of the planets are placed on a volume envelope in the form of a parabolic along the Sun s trajectory. - The planet is born from the active Sun billions year ago by a small-bang.(t=0 ; r=zo ) - The moons are born from the active Sun-Planet also billions year ago by small-bangs. - Planet and its moons are in equilibrium inside their family system and acts as single mass. - Period law is not valid for Kepler s orbits, or for spiraled orbits. - Newton says: period law is valid for circular orbits with uniform peripheral velocity, - Kepler says: period law is valid also for elliptical orbit and accelerated motion on the orbit - Spiral theory says: as (r) and time are variables, there is not a constant period for the planets. Period definition is not valid but real time (t) should be considered for the comparison of (r). This is (r1/r2)=(vp1/vp2)^2. 6. CRACKING THE COMMENTS According Kepler, the planets were at their actual position since the beginning and will stay on these elliptical orbits for eternity, on a mathematically blocked, rigid mathematical ellipse.spiraled theory says: the planets are born from the active Sun with a small bang, their distance to the Sun is changing every second, their rotation around the Sun is obligatory a spiral and this rotation time is changing for each cycle. There is no period, but real time for one rotation. Period is a repeating time for repeating motion. There are not repeating celestial motions. As there is only one extreme (r) for the total life-time of the planet, Aphelion and Perihelion definitions are not valid. There is not a constant distances repetition for each cycle of the planet around the Sun. All distances start from zero, go to a maximum and then return back to zero. Like for a parabola on Cartesian, with Vx=Ct and VyMax=(2*a*rMax)^(1/2). Table I, shows the relative time, real time, special dates, formulary distance, corresponding distance, period variations, days in a cycle, for the planet Earth, according the data of era 2009. Table I 244
7. WHY DO THE PLANETS ORBIT THE SUN? When we say Fattraction=Fcentripetal, we mean equilibrium: G*m1*m2/d^2=m1*Vt1^2/r1=m2*Vt2^2/r2 Vt1 and Vt2 must exist for the equilibrium. This must make the planets orbit while attracting each other. Attraction is not linear due to Vt, but spiraled obligatory. Planet and their moons are considered as a family represented by (m1).fig 5 Fig 5: Obligatory orbits, where r*vp^2=ct for a given era is valid instead of period law. 8. EASY CONFIRMATION: (TRY IT TO BELIEVE) Have the data of the planets for era 2009. You find for all the planets, even for Halley and Pluto VpEarth^2*rEarth=(29,786 km/sec)^2*(149597890 km)=132 724 771 939,26 km^3/sec^2 VpMars^2*rMars=(24,131km/sec)^2*(227939150 km)=132 724 771 939,26 km^3/sec^2 Newton s period law never existed for celestial orbits, while it is correct for mechanical conditions. Kepler discovered the period laws in 1618 and used it for non-circular and accelerated motion. While, 50 years after Kepler, Newton proved that period law is valid only and only for circular movement with constant peripheral velocity. This is a discrepancy, which should be corrected today. In his book PRINCIPIA Newton researched to prove Kepler s discoveries. He did not confirm Kepler. But he found that celestial orbits should be spiraled. Unfortunately he refused spiraled orbit, due to his period law, thinking that on such orbits celestial bodies will go on ad infinitum.[1] As the bodies are at our reach for all evaluation, their orbit should not be spiraled, or hyperbolic or parabolic, he said. 245
But planets do not go on ad infinitum. They are located in finitum.[1] r= -4*t^2+4*t*T-4*T^2/6 is the sign of finite location. And (F attraction=f centripetal) means spirals. Due to the existence of Vp, the bodies do not attract each other like magnets on linear trajectories but on spiraled trajectories. No ellipse, anywhere. Fig.6 explains how spirals are formed around the Sun. Bodies are ejected from the Sun; even today But today's Sun is too weak for an expulsion of large bodies. Fig 6: Spirals around the Sun: with Parabolic steps in expansion and in contraction. There is only one extreme. Fig.7 is about a 3D view of the orbits from a galaxy. Spirals are enveloped by a Parabolic volume. Fig 7: Orbits seen from a galaxy outside of the Milky-way 246
9. CONCLUSİON Even these considerations necessitate the correction of our educational system. Clearly, it is difficult to agree with novelties, when the community is trained on wrong direction since 400 years. REFERENCES [1] Newton s PRINCIPIA, by A.MOTTE.page290 Author Profile NecatTaşdelen is Diplomedmech. Engineer; retired. In 1959 discovered the formula (a^s+b^s=l^s), a Thales theorem result, for the estimation of elliptical curves perimeter and similar. For an ellipse, the accuracy of the evaluations was error %=0,000002...for the whole range: [1<(b/a)<infinity].The most accurate of that time; even today. 247