Related: [[Geocentrism]] [[Attachments/Database Tabs 2/Michelson Morley|Michelson Morley]] [[Attachments/Database Tabs 2/Michelson-Gale Experiment explained 1|Michelson-Gale Experiment explained 1]] [[Library/Database Tabs/Aether Wind Detection Tests 1|Aether Wind Detection Tests 1]] kinematics - abstract math used to analyze motion in a system Dynamics - using real forces to try to predict motion in a system ![[Attachments/Pasted image 20240813173906.png]] **Kepler's Laws** are introduced with a foundation in elliptical geometry, describing the orbital paths of planets around the Sun, which serves as one of the foci of the ellipse. The three laws are: 1. **First Law**: Planets orbit the Sun in ellipses with the Sun at one focal point. 2. **Second Law**: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time, leading to varying orbital speeds. 3. **Third Law**: The square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit. The proportionality between the squares of the orbital periods and the cubes of the semi-major axes of orbits (Kepler's third law) can be derived from the gravitational interactions between celestial bodies ![[Attachments/Pasted image 20240403081523.png]] ![[Attachments/Pasted image 20240403081531.png]] ![[Attachments/Pasted image 20240403081542.png]] ![[Attachments/Pasted image 20240403081552.png]] ![[Attachments/Pasted image 20240408135944.png]] ![[Attachments/Pasted image 20240408140123.png]] ![[Attachments/Pasted image 20240409151607.png]] ![[Attachments/Pasted image 20240409151644.png]] When Michael Brenner Exposed the Equivalence Principle https://www.youtube.com/watch?v=l7MAOk46DuQ&ab_channel=AetherCosmology Kepler’s laws are purely empirical and, when stated, has no theoretical backing. It is interesting that Kepler reached all these conclusions from data gathered with the naked eye (primarily data that was collected by Tycho Brahe (1546–1601) since his investigations took place before the invention of the telescope by Galileo (1564– 1642) in 1610. Kepler’s laws were simply observations, however. Therefore, one of the biggest questions of the Renaissance was: “Why (or perhaps more accurately, “how”) do the planets obey Kepler’s laws?” Sir Isaac Newton ![[Attachments/Pasted image 20240128124205 1 1 1 1 1 1 1.png]] ![[Attachments/Pasted image 20240329085753.png]] ![[Attachments/Pasted image 20240311072501.png]] Robert Bennet: The Kepler's Third law says that the square of the period of the planets is proportional to the cube of the radius. This was discovered pretty long time ago, almost 400 years, more than 400 years. You can actually derive Kepler's third law from Newton's law. The force of gravity on any planet. This mass of the planet times its radius times its angular velocity squared. The angle of velocity is two pi over t, where t is the period. So you can derive Kepler's law from Newton's law. B<font color="#00b0f0">ut notice that the mass of the planet disappears. It cancels out.</font> And this is characteristic of when you have something that is kinematic, something that depends only on empirical observation and doesn't depend on a law of physics. Deriving orbits with Kepler's laws and Newton's constants turns out to be physically meaningless kinematic equations after all. G and M give you a proportionality to the periodicity of the observation. Meaning these equations don't actually account mass attracting mass. At least not in any meaningful way like we've been led to believe. Electricity voltage is pressure. how hard its being pushed amperage how much water ![[Attachments/Pasted image 20240504015833.png]] Deriving Kepler's 3rd law from Newton's law. Notice the Mass canceling out, i.e. it was just a kinematic equivalence with no physical meaning. ![[Attachments/Pasted image 20240311072511.png]] ![[Attachments/Pasted image 20240311072514.png]] ![[Attachments/Pasted image 20240316190818.png]] Derivation of Kepler's Third Law from Newton's Laws 1. Newton's Second Law Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: =—F=m—a 2. Gravitational Force According to Newton's law of universal gravitation, the gravitational force between two objects with masses 1m1 and 2m2 separated by a distance r is given by: =—1—22F=r2G—m1—m2 3. Centripetal Force For an object of mass m orbiting a central body of mass M, the centripetal force is provided by gravitational force: =—2=——2F=rm—v2=r2G—M—m 4. Equating Centripetal Force and Gravitational Force Equate the expressions for centripetal force and gravitational force: —2=——2rm—v2=r2G—M—m 5. Solving for Orbital Acceleration a Using Kepler's second law, which states that the area swept out by the radius vector of a planet is constant with time, for a circular orbit: =2=—(2/)2=242—a=rv2=rr—(2r/T)2=42—rT2 6. Substituting a into the Equation 242—=——242—rT2=r2G—M—m 7. Solving for r and T Rearrange the equation to solve for r: 3=42———21r3=142—G—M—T2 8. Conclusion Comparing the derived equation with Kepler's third law, we see that they are in the same form, confirming that Kepler's third law can be derived from Newton's laws of motion and gravitation. ![[Attachments/Pasted image 20240316021014.png]] ## To derive while canceling out the mass. 1. ** Start with the equation for centripetal force and gravitational force:** Centripetal force=Gravitational forceCentripetal force=Gravitational force —2=—(—)2rm—v2=r2G—(m—M) 2. **Express the orbital velocity v in terms of the orbital period T and the radius r:** From Kepler's second law, we know that the area swept by the radius vector per unit time is constant, which leads to =2v=T2r. 3. **Substitute v into the equation for centripetal force:** —(2)2=—(—)2rm—(T2r)2=r2G—(m—M) 4. **Solve for G:** =42—32—G=T2—M42—r3 5. **Compare G with the expression for the gravitational constant in Newton's law:** In Newton's law, =—21—2G=m1—m2F—r2. 6. **Substitute the expression for G derived from Kepler's laws into Newton's law:** =(—)—42—32——12F=T2—M(m—M)—42—r3—r21 =——422F=T2m—M—42 This shows that by canceling out the mass m in both the centripetal and gravitational force equations, we end up with Newton's law of universal gravitation directly from Kepler's laws. In this derivation, you can see that the mass m cancels out in the final equation, indicating that the force F is independent of the mass of the satellite. This phenomenon is characteristic of a kinematic relationship rather than a dynamic one involving mass attracting mass. It suggests that the observed orbital motion described by Kepler's laws is not fundamentally based on the gravitational interaction between masses but rather on empirical observations of planetary motion. This realization raises questions about the true nature of the relationship between mass and gravity and challenges the traditional understanding of gravitational theory. ![[Attachments/Pasted image 20240316020002.png]] ![[Attachments/Pasted image 20240316020940.png]] ![[Attachments/Pasted image 20240406020052.png]] ![[Attachments/Pasted image 20240406020113.png]] ![[Attachments/Pasted image 20240406020023.png]] ### Centripetal Force and Its Relationship to Orbital Motion **Centripetal Force** is the force required to keep an object moving in a circular path. It acts towards the center of the circle, constantly changing the direction of the object's velocity to keep it in circular motion. The formula for centripetal force (FcF_cFc​) is: Fc=mv2rF_c = \frac{mv^2}{r}Fc​=rmv2​ where: - m is the mass of the object, - v is the tangential velocity of the object, - r is the radius of the circular path. ### Relationship to Orbital Motion In the context of planetary motion, the centripetal force is provided by the gravitational force between the planet and the Sun. This gravitational force keeps the planet in its elliptical orbit. **Newton's Law of Universal Gravitation** states that the gravitational force (FgF_gFg​) between two masses (m1m_1m1​ and m2m_2m2​) separated by a distance rrr is: Fg=Gm1m2r2F_g = \frac{G m_1 m_2}{r^2}Fg​=r2Gm1​m2​​ where: - GGG is the gravitational constant, - m1m_1m1​ is the mass of the Sun (or central body), - m2m_2m2​ is the mass of the planet (or orbiting body). For a planet of mass mmm orbiting the Sun at a distance rrr, the gravitational force provides the necessary centripetal force to maintain the orbit: GMmr2=mv2r\frac{G M m}{r^2} = \frac{mv^2}{r}r2GMm​=rmv2​ where MMM is the mass of the Sun. ### Deriving Kepler's Third Law from Centripetal and Gravitational Forces Kepler's Third Law states that the square of the orbital period (TTT) of a planet is proportional to the cube of the semi-major axis (aaa) of its orbit: T2∝a3T^2 \propto a^3T2∝a3 To derive this, we start by equating the centripetal force and gravitational force: GMmr2=mv2r\frac{G M m}{r^2} = \frac{mv^2}{r}r2GMm​=rmv2​ Solving for vvv: v2=GMrv^2 = \frac{G M}{r}v2=rGM​ The orbital period TTT is the time it takes for the planet to complete one orbit, which is the circumference of the orbit divided by the velocity: T=2πrvT = \frac{2\pi r}{v}T=v2πr​ Substituting vvv from the previous equation: T=2πrGMrT = \frac{2\pi r}{\sqrt{\frac{G M}{r}}}T=rGM​​2πr​ T=2πr3GMT = 2\pi \sqrt{\frac{r^3}{G M}}T=2πGMr3​​ Since the semi-major axis aaa is equivalent to rrr for a circular orbit: T2=(2π1GM)2r3T^2 = \left( 2\pi \sqrt{\frac{1}{G M}} \right)^2 r^3T2=(2πGM1​​)2r3 T2=(2π1GM)2a3T^2 = \left( 2\pi \sqrt{\frac{1}{G M}} \right)^2 a^3T2=(2πGM1​​)2a3 This shows that T2T^2T2 is proportional to a3a^3a3, confirming Kepler's Third Law. ### Summary - **Centripetal Force**: Required to keep an object in circular motion, provided by gravitational force in the context of planetary orbits. - **Gravitational Force**: The attractive force between two masses, which provides the necessary centripetal force for orbital motion. - **Kepler's Third Law**: Derived by equating gravitational force and centripetal force, showing the relationship between the orbital period and the semi-major axis of an orbit. In the images provided, the equations and concepts discussed above are visualized, showing how Newton's laws of motion and gravitation confirm Kepler's empirical laws of planetary motion. ![[Attachments/Pasted image 20240705222426.png]] ![[Attachments/Pasted image 20240705222707.png]] ![[Attachments/Pasted image 20240705222725.png]] ![[Attachments/Pasted image 20240705222822.png]] Transcript of Alan describing this in Aethercosmology video: https://www.youtube.com/watch?v=WllD_OQzOyc&ab_channel=AetherCosmology   📍 All right. We have periodicity, which is just when an event happens with a specific time interval. An example of that would be how long does it take the sun to complete a circuit or for heliocentrists? How long does it take for the earth to revolve around the sun? That kind of thing. All right. You're just making measurements and you're making assumptions about the motion and you can't really do any testing in any meaningful way, so you're just making measurements. That gives you a periodicity. All right, so as we move forward, we're going to talk about Kepler's laws, but before we do that, we're going to get some prerequisite ellipsoid knowledge or ellipse knowledge. So an ellipse is defined by a focus, point, or a foci when you combine the two. So these are two equidistant points that will remain constant no matter any point on the circuit of the ellipse. And then the flattening of an ellipse will be defined by a number between 0 and 1, where 0 is a perfect circle and 1 would be. A straight line, so that would be like the deviation from that. And then you can break the ellipse up into 3 axes. The longest access between them would be. The horizontal 1 right here, this would be your major access. The shortest access vertically would be the minor access and then the half of the major access would be the semi major access. So you can break it up into three parts right there. And then we're going to get into Kepler's laws as soon as this slide transitions. .So the sun will be in this model, the sun will always be a focus point here, and then the planets will orbit around that focus point and that will create their elliptical orbit. Okay, so that's the, difference between the 2 models here. So the geocentric model here, and then the heliocentric model with Kepler's elliptical orbits. So the deviation from the perfect circle is is his model, right? So now we're going to discuss the laws that he puts forward. So if Kepler's 1st law, the orbit of a planet is an ellipse. With the sun at one of the 2 focal points, right? Or focal foci points, right? So we discussed how the ellipse works, the geometry of it, where the sun would be, and then the planets, how they orbit around it. Here's a refresher on that mechanism here. We got this, we got the sun and we got the planet and that's, it's always going to be at a focus point, right? That'll be where the sun's at now. For the 2nd law, we have a planet in its orbit sweeps out. Equal areas with time are in equal time. Planets move faster when it's in perihelion and slower when it's aphelion. So to give a visual of that, we have a slide from man of stones video that we're going to be touching on towards the end of the presentation here. ![[Attachments/Pasted image 20240408135215.png]] So when a body is perihelion, this is when it's going to be moving fastest. And when it's furthest away from the sun, that's appealing. And that's when it's going to be moving at slowest. Yeah. Kepler's third law is where they, apply these, This knowledge to get the the ratio for the, for distances and stuff, right? So this is how they apply distances to the sky with kinematics, right? So the squares of the orbital periods of the planets are directly proportional to the cubes of the semi major axis of their orbits, right? And we'll get into what all that means specifically as we move forward, but that's those are, the laws there. And this gives way to the astronomical unit, which will serve as the baseline for measuring everything out in the cosmos and all that. Quick refresher. Everything we went over was kinematics. That was all measurements of motion, irrespective to causal mechanisms. Moving forward, we will get into the dynamics and we'll see an attempt to apply force in a causal mechanism to their motion. If a relationship can be established, it would be good. It would be in good agreement with the theory for the causal mechanism mathematically, right? If you could show a relationship between the distance derived from the proportionality of the periodicity. And then in relation to an actual cause of the motion, that's also produces that same distance relationship. Sorry. Okay. Let's see. So that would act as the gravitational force between two bodies and that would produce a centripetal force that would be equivalent to the gravitational force. So we're going to represent Kepler's laws with actual dynamics here, right? This is like the, this is how they're going to merge these two, right? Because they say that you can derive Kepler's laws from Newton's and vice versa, meaning they have a covariant relationship so that thing we talked about earlier, how it would, be nice if you could establish a relationship between the periodicity and the forceful causal mechanism, right? That would be good evidence that your theory has a good dynamics, right? Let's see here. I think that's everything for that. Okay, cool. So what we just learned was we were going to apply centripetal force to being equivalent to gravitation. And that breaks down to GMM over four pi R squared times. T squared is equal to the square like, is equal to the semi major axis cube, right? So this is Kepler's law right here. And then this is Newton's law. And we're going to use these to derive 1 another and show the so show that they're equivalent. So that was the whole purpose of, what they did or what they established there. With the cavendish experiment, right? They establish a constant for gravity. So what they're going to do, they're going to start off. With earth of how they model out the whole solar system, right? And then in that experiment, they also derive the constant for M2, right? So they take that relationship. They take the, oh, by the way, real quick. Sorry, I got a little carried away here. Doing a little algebra and canceling everything out to make sure everything's all equivalent on the up and up to get the reduced form of this. We're looking at the we're, looking at the finalized form essentially. So we have to, convert the centripetal force. To distance over time, which they do, which is done with this 4, 4 pi R squared, which yeah, you're just missing an R in there. That's all. Yeah. Sorry. That's what was throwing me off. Okay. What that does is that gives them that changes this velocity here. How fast something's moving into the sky. This is how they're going to relate it to an orbital distance, right? This is going to give them the circumference of that. They've got that. Okay. And this is the, this is what the, this is the conversion for the velocity, right? So now we're going to take our 2 constants from the experiment from Cavendish that they established, and we're going to apply a periodicity, right? 365 days for the sun to complete a circuit or for the earth to complete a, A revolution around the sun, right? So this is what we're looking at, right? And then when they apply all that, they get a distance, right? That's proportional to the cube of the semi major axis, right? So that's where they establish their baseline for 93M miles. And because the mass in this equation cancels out, and it's just about these constants in the periodicity of the event, you would get the results. It doesn't matter, right? It's going to, it's going to establish a proportional ratio anyway, because it's about the periodicity. All right. So are you saying the causal mechanism, the dynamic force that's being put forward as mass attracting mass isn't substantiated in these equations that's put forward? Because what will happen? Are you saying that the constants cancel out and then you're only left with the periodicity? No the mass of the 2 bodies, so it's just about the periodicity of the event. And then they could backwards derive the mass of something. For example, once they establish this relationship, they establish the mass of the sun, and then they go, okay, we're going to put the big M here, the G here, and then we're going to say, what about Mars? Okay. The periodicity of Mars as it goes around. Okay. It completed its thing. And then now it's proportional to this distance right here. So that's how they do it all and then they go, okay, how about your boy Jupiter? Right now? We know Jupiter's angular size and all that, but you can't tell anything about that. But when you apply these equations with the mass of the sun, gravitational influence of this body here in relation to the periodicity, it's going to give you the orbital. The semi major axis orbit of it, right? So it's going to give you the distance of it. Sounds like kinematics to me. Since the causal mechanism is canceled out, and it's about the period is to the, of the event to scale the proportionality. I would say that's an accurate assessment that's a cool way to do that. That was a good idea to just do it over each orbital body like that. Thank you. Yeah, I was going to mention that to Toby, because we've all seen the solar system in an image, and we've heard all about the math, but, you Having to move it around physically and just talking it through is like the mathematical model fits the what they show us the model. Hey, Alan, can you go back a frame just for a sec? Just for mathematical clarity here? Because Shane was touching on it. The M, the small M there that's in the GMM is the thing that is being canceled out on the other side of the equal sign up top against the M V squared. That small M there and the small M in the GMM, we're dividing both sides by M and that's how they cancel them out. They, divide by the same value. And all of a sudden, there's no 2nd mass on either side. Anyways that's, the interesting trick to pull on the only way they can determine mass in the first place is by applying this question. So it's do any of them have really mass in reality? And the answer is, of course no, not at all. But like someone was just pointing out, though, once you solve that, you can backwards derive what the mass would be. So when, so people will say, okay, how come when I apply Newton's equations to Jupiter's moons, I can derive the periodicity based on Jupiter's mass and all that. It's it's did you follow the steps of getting the Jupiter's mass off of the ratio of the sun? Like following Kepler's laws, because if so, it's going to output whatever that moon needs to be to fit that orbital period into like that. That's what it's going to output. So crazy, it reminds me of the three body problem. They're like, yeah, we can't solve it. So what we're going to do is break it up into individual two series, probably change all the parameters and do the sun and each body and then put it all together at the end. Like we solved it. Yep. And then superimpose it all together and actually it's a general solution to this guy. And thank you stone for that clarification on the canceling out. That is super important on the mechanism of the cancel out. Much appreciated. Okay, so this slide got messed up, but I'm going to go ahead and just read the notes as is these equations are supposed to represent the dynamics of the solar system. But do they truly represent the dynamics of the solar system? the application of Kepler's laws and Newton's law of gravitation to model the dynamics of planetary orbits, with a focus on the importance of mass and the mathematical relationships that describe these dynamics. It starts with an introduction to periodicity, the concept of regular occurrences at specific intervals, as exemplified by celestial bodies' movements. It then transitions into a discussion of ellipses, which are central to understanding Kepler's laws of planetary motion, particularly how planets orbit the sun in elliptical paths with the sun at one focal point. **Third Law**: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically, 2∝3T2∝a3, where T is the orbital period and a is the semi-major axis. ![[Attachments/Pasted image 20240408135438.png]] ![[Attachments/Pasted image 20240408135455.png]] ![[Attachments/dfsdfsfdsdsfsd.png]] ![[Attachments/photo_2023-05-08_14-48-34.jpg]] https://www.youtube.com/watch?v=CCsbSq9wlyI&ab_channel=TheOrganicChemistryTutor ![[Attachments/Pasted image 20240311072516.png]] ![[Attachments/Pasted image 20240127193546 1 1 1 1 1 1 1.png]] https://www.youtube.com/live/5j0wCrkzfwk?si=IKJDYt0Uf0cRH-Px&t=3239 (2019). "The Derivation of Kepler’s Laws of Planetary Motion From Newton’s Law of Gravity." ![[Attachments/Pasted image 20240311072521.png]] ![[Attachments/Pasted image 20240311072524.png]] ![[Attachments/Pasted image 20240311072526.png]] ![[Attachments/Pasted image 20240204011659 1 1 1 1 1 1.png]] ![[Attachments/Kinematics and Dynamics Kepler from Newton.png]] ![[Attachments/Pasted image 20240311072532.png]] ![[Attachments/Pasted image 20240311072539.png]] ![[Attachments/Pasted image 20240311072542.png]] ![[Attachments/Pasted image 20240311072545.png]] ![[Attachments/Pasted image 20240311072549.png]] ![[Attachments/Pasted image 20240204011913 1 1 1 1 1 1 1.png]] ![[Attachments/Pasted image 20240204011922 1 1 1 1 1 1 1.png]] ![[Attachments/Pasted image 20240204011934 1 1 1 1 1 1 1 1 1 1 1 1 1.png]] ![[Attachments/Pasted image 20240315232515.png]] Newton was able to propose more general laws that describe the motion of an object under the influence of any force, but in particular the force of gravity. Read about them by next class, but it may help if you keep in mind why you are reading about this: Newton’s laws will give us a way, basically our only way, to get the masses of objects, first stars that orbit each other (binary stars), then a technique to detect black holes, since 1995 the masses of extrasolar planets, and the evidence that there is some invisible mass called “dark matter.” Then try to answer this apparently boring question: Gravity is what makes objects orbit around other objects, and gravity is a reflection of an object’s mass. So why doesn’t the mass of the objects appear in Kepler’s 3rd law? Newton’s laws of motion and gravi Every body continues in a state of rest or uniform motion (constant velocity) in a straight line unless acted on by a force. (A deeper statement of this law is that momentum (mass x velocity) is a conserved quantity in our world, for unknown reasons.) This tendency to keep moving or keep still is called “inertia.” 2. Acceleration (change in speed or direction) of object is proportional to: applied force F divided by the mass of the object m i.e. ! a = F/m or (more usual) F = ma This law allows you to calculate the motion of an object, if you know the force acting on it. This is how we calculate the motions of objects in physics and astronomy. ! You can see that if you know the mass of something, and the force that is acting on it, you can calculate its rate of change of velocity, so you can find its velocity, and hence position, as a function of time. 3. To every action, there is an equal and opposite reaction, i.e. forces are mutual. A more useful equivalent statement is that interacting objects exchange momentum through equal and opposite forces. Hey explain how this is not a problem for you or admit you have no idea what believe you please This would only happen when things are kinematic, and depend only on empirical observation and doesnt follow any law of physics. alright. stay with me here ... When you derive Kepler's third law from Newton's law, the mass cancels out. And this is characteristic of when you have something that is kinematic, something that depends only on empirical observation and doesn't depend on a law of physics. Thats why. Deriving orbits with Kepler's laws and Newton's constants turns out to be physically meaningless kinematic equations after all. G and M give you a proportionality to the periodicity of the observation. Meaning these equations don't actually account for mass attracting mass AT ALL! At least not in any meaningful way like we've been led to believe. Kepler's laws, derived from empirical data without a theoretical basis, describe [planetary motion](https://en.wikipedia.org/?curid=17553), while Newton's law of gravity provides a theoretical explanation for the motion of objects under the influence of gravity. The paper shows that Kepler's third law, which states that the square of the period of planets is proportional to the cube of the radius, can be derived from Newton's law, but interestingly, the mass of the planet disappears in this derivation, leading to the conclusion that it is kinematically derived and lacks physical meaning. ## All Orbits are Conic Sections ![[Attachments/Pasted image 20240507212456.png]] ![[Attachments/Pasted image 20240507212635.png]] ![[Attachments/Pasted image 20240507212640.png]] ![[Attachments/Pasted image 20240507212624.png]] ![[Attachments/Pasted image 20240409152030.png]] Other Derivations ![[Attachments/Pasted image 20240409152153.png]] ![[Attachments/Pasted image 20240409152143.png]] This has NOTHING to do with the dynamic cause of gravity as mass attracting mass..... a = F/m or (more usual) F = ma This law allows you to calculate the motion of an object, if you know the force acting on it. This is how we calculate the motions of objects in physics and astronomy. ! You can see that if you know the mass of something, and the force that is acting on it, you can calculate its rate of change of velocity, so you can find its velocity, and hence position, as a function of time. ![[Attachments/MathHistory-22.pdf]] https://www.cas.miamioh.edu/~alexansg/phy111/lectures/05_kepler/kepler.pdf https://mast.queensu.ca/~murty/MathHistory-22.pdf https://www.math.ksu.edu/~dbski/writings/planetary.**pdf** ![[Attachments/planetary.pdf]] ## Videos: Kinematics vs Dynamics: Physically Meaningless Equations Used to Describe Celestial Motions [https://www.youtube.com/watch?v=xK4YEki9P8s](https://www.youtube.com/watch?v=xK4YEki9P8s "https://www.youtube.com/watch?v=xK4YEki9P8s") Geocentric vs Heliocentric Parallax Angle Predictions https://www.youtube.com/watch?v=AjGF7K3tEPA How Math Became Reality https://www.youtube.com/watch?v=D7uXY_AuWLk https://www.atnf.csiro.au/outreach//education/senior/astrophysics/binary_mass.html ## Deriving Equations for Mass of Binary System The barycenter or centre of mass of the system is where: mArA = mBrB (Equation 5.1) and as r =rA + rB (5.2) then rB = r - rA so mArA = mB(r - rA) ∴ rA = mBr/(mA + mB) or rA = mBr/M (5.3) where M is the total system mass. The forces acting on each star are balanced, that is the gravitational force equals the centripetal force so; FG = FC or GmAmB/r2 = mAv2/rA (5.4) where v is the orbital speed of A. Unless v can be measured or inferred directly from Doppler shift in its spectrum it must be calculated from the period, T: v = 2πrA/T so substituting this into (5.4) gives: GmB/r2 = 4π2rA/T2 so if we then substitute in (5.3) we get: GmB/r2 = 4π2mBr/T2M or: M = 4π2r3/GT2 (5.5) which can be rewritten as: **mA + mB = 4π2r3/GT2** (5.6) (This is the form specified in the HSC formula sheet) now (5.5) is simply an expression of Kepler's 3rd Law; r3/T2 = GM/4π2 (5.7) Using equation 5.5 or 5.6 we can determine the mass of the binary system if we can measure the orbital period and the radius vector (separation between the two components) for the system. In practice most systems will not have their orbital plane perpendicular to us so we need to adjust for the observed inclination. Whilst it is relatively straight forward to determine the total system mass, it is harder to determine the individual masses of the component stars. This requires the distance from a component star to the barycenter to also be measured. We can then use this to determine the mass of that star by using: mA = M(r - rA)/r (5.8) Once the mass of one component and the total system is known it is straightforward to calculate the mass of the other component. ## Examples of Mass Calculation in Binaries **Example 1: Determining the total mass of a system**. The α Centauri system is 1.338 pc distant with a period of 79.92 years. The A and B components have a mean separation of 23.7 AU (although the orbits are highly elliptical). _What is the total mass of the system?_ To solve this we use equation 5.5 but before we simply substitute in we need to check units. Whilst we can use use AUs and years to give us a relative value when applying Kepler's Laws, if we want a full numerical solution we must convert all parameters to S.I. units as the constant G is normally expressed in such units (G = 6.672 × 10-11m3.kg-1.s-2). so T = 79.92 yrs = 79.92 × 365.25 × 24 × 60 × 60 = 2.522 &times 109 s, and r = 23.7 AU = 23.7 × 1.50 × 1011 m = 3.55 × 1012 m. Now substituting these into (5.5); M = 4π2r3/GT2 gives: M = 4π2(3.55 × 1012)3/((6.672 × 10-11) ×(2.522 &times 109) 2) **∴ M = 4.162 × 1030 kg**. **Example 2: Calculating the mass of one of the component stars**. As α Cen is a nearby visual binary system, careful astrometric observations reveal that the primary component, α Cen A has a mean distance of 11.2 AU from the system's barycenter. _What is the mass of each of the component stars in the system?_ From example 1 above we have already calculated the system mass to be 4.162 × 1030 kg. Now distance rA = 11.2 AU = 11.2 × 1.50 × 1011 = 1.680 × 1012 m. We shall use this to first find the mass of α Cen A. Using equation (5.8): mA = M(r - rA)/r and substituting in: mA = 4.162 × 1030(3.55 × 1012 - 1.680 × 1012)/3.55 × 1012 ∴ mA = 2.192 × 1030 kg. **so α Cen A has a mass of 2.192 × 1030 kg**. Now to find mass of α Cen B we simply use M = mA + mB so that: mB = M - mA mB = 4.162 × 1030 - 2.192 × 1030 ∴ mB = 1.970 × 1030 kg. **so α Cen B has a mass of 1.970 × 1030 kg**. The mass of α Cen A makes it about 1.1 × that of our Sun. It, too is a G2 V star with a luminosity about 55% greater than that of our Sun. α Cen B is a dimmer K0-1 V min sequence dwarf with 90% of the Sun's mass and only about half as luminous.  This image shows the orbits of an L-type star and its brown dwarf companion. The period is about ten years. Using the data from these images astronomers measured the mass of the L-type star to be about 8.5% that of our Sun and the brown dwarf to be 6.6%. The individual snapshots of the orbit were obtained using the HST and adaptive optics on the VLT, Keck and Gemini. For more detail visit the [press release](http://www.eso.org/outreach/press-rel/pr-2004/pr-16-04.html#phot-19a-04). Of particular value to astronomers are systems that are both eclipsing and spectroscopic. The radial velocity values from the spectral data can be used to calculate absolute rather than just relative values for the stellar radii. This can then be combined with orbital inclination parameters obtained from the light curve to give the stellar masses and mean stellar densities. The relative luminosities and total luminosity of the system can be derived then used to calculate the total flux of the system. This then allows us to calculate the distance to the system. We can also infer the mass and luminosity of each star. As already mentioned, binary stars are of vital importance as they allow us to determine stellar masses. To date (August 2004) only one single star other than our Sun has had its mass accurately determined by a means unrelated to Kepler's laws. Astronomers involved in the MACHO project at Mt Stromlo took an image in 1993 that showed microlensing of a distant background star by a closer star in the foreground. Recent parallax measurements and observations by the HST have allowed astronomers to calculate that the faint, red foreground star has a mass only one-tenth that of our Sun. For more information, read the [press release](http://hubblesite.org/newscenter/newsdesk/archive/releases/2004/24/text/).  1. **Inverse Square Law**: Newton's Law of Gravitation is an inverse square law, meaning the force between two masses is inversely proportional to the square of the distance between them. 2. **Conic Sections**: Under the influence of an inverse square force, the orbit of a body around another one will trace out a conic section, which can be a circle, ellipse, parabola, or hyperbola, depending on the energy and angular momentum of the system. 3. **Derivation**: The proof provided in your statement involves the use of polar coordinates (r, θ) where the potential V is central, depending only on the distance r. Using the substitution u = 1/r simplifies the equation of motion into a form that reveals the shape of the orbit. 4. **Differential Equation**: The second-order differential equation derived from these principles describes the motion of the body in orbit. Its general solution shows that u (and hence r) varies with θ in a sinusoidal manner, consistent with conic sections. 5. **Energy and Orbit Shape**: The constant E represents the total mechanical energy of the system. The value of E determines the shape of the orbit: - If E < 0, the energy is negative, and the orbit is elliptical (bound orbit). - If E = 0, the orbit is parabolic (marginally bound). - If E > 0, the energy is positive, and the orbit is hyperbolic (unbound). 6. **Orbit Equation**: The orbit equation given relates r (the radial distance) to θ (the angular position), with ε representing the eccentricity of the conic section. 7. **Special Case (Circular Orbit)**: If ����=0dθdu​=0, it indicates that the orbit is circular, as the radial distance r does not change with θ, and hence the body is at a constant distance from the focus of the conic section. ![[Attachments/Pasted image 20240409151503.png]] This theorem and its proof are fundamental in celestial mechanics and have been validated by centuries of observation and analysis. It indeed shows how Kepler's laws, which were empirical, can be derived from Newton's more fundamental gravitational theory, establishing the connection between geometry (conic sections) and physics (forces and motion). ![[Attachments/Keplers-Laws (1).pdf]] https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/13%3A_Gravitation/13.06%3A_Kepler's_Laws_of_Planetary_Motion ![[Attachments/13.06__Kepler's_Laws_of_Planetary_Motion.pdf]] https://web.stanford.edu/class/math63cm/kepler.pdf https://www.physicsclassroom.com/getattachment/Physics-Video-Tutorial/Circular-Motion-and-Gravitation/Keplers-Three-Laws/Lecture-Notes/LessonNotes.pdf?lang=en-US ![[Attachments/LessonNotes.pdf]] ![[Attachments/kepler.pdf]] ![[Attachments/planetary (1).pdf]] Yes. Here is the derivation in Goldstein (p101) for example... ![[Attachments/Pasted image 20240706011821.png]] well, changing the constant k to try to shove it back in doesn't QUITE work..... right? ![[Attachments/Pasted image 20240706011807.png]] ![[Attachments/GVHN3-gWYAMV3R-.png]] ![[Attachments/Pasted image 20240816112728.png]] The image depicts a problem related to the gravitational interaction between two masses in a two-body system, likely within the context of orbital mechanics or classical mechanics. Here's a breakdown of the key concepts illustrated: ### **1. Center of Mass:** - The equation at the top indicates the position of the center of mass for two bodies with masses \( m_1 \) and \( m_2 \). - \( r_1 \) and \( r_2 \) represent the distances of each mass from the center of mass. - The relationship is given by \( r_1 m_1 = r_2 m_2 \), with the total distance \( r_1 + r_2 = d \). ### **2. Gravitational Forces:** - The gravitational forces between the two masses \( m_1 \) and \( m_2 \) are calculated using Newton's law of universal gravitation: \[ F = \frac{G m_1 m_2}{d^2} \] where \( G \) is the gravitational constant, and \( d \) is the distance between the two masses. ### **3. Orbital Period:** - The lower part of the image shows a derivation of the orbital period \( T \) for the system. - The formula presented seems to conclude with: \[ T = 2\pi \sqrt{\frac{d^3}{G(m_1 + m_2)}} \] This is Kepler's third law, applied to a two-body system, where \( T \) is the orbital period, and \( d \) is the semi-major axis of the orbit. ### **Summary:** This diagram and derivation seem to provide a concise analysis of the gravitational interaction and orbital dynamics of a two-body system, likely leading to a derivation of the orbital period in terms of the system's total mass and separation distance. If you have any specific questions about this derivation or need further clarification on any part of it, feel free to ask!