m1---> <---m2 F12 F21 Figure 1 - Example 1: 2 objects |
F12 = F21 = G * (m1*m2)/r^2 | (Equation 1) |
F12 = m1 * a1 = G * (m1*m2)/r^2 F21 = m2 * a2 = G * (m1*m2)/r^2 |
(Equation 2a) (Equation 2b) |
F12 = a1 = m2/r^2 F21 = a2 = m1/r^2 |
(Equation 3a) (Equation 3b) |
^ | v2 | <---m1---> C <---m2---> | F12 F21 v1 | V <--------r1--------><--------r2-------> Figure 2 - 2 objects |
a1 = m2/r^2 = v1^2/r1 a2 = m1/r^2 = v2^2/r2 |
(Equation 4a) (Equation 4b) |
m1 * a1 = (m1*m2)/r^2 + (m1*m3)/r^2 | (Equation 5) |
a1 = m2/r^2 + m3/r^2 | (Equation 6) |
m2 * a2 = (m2*m1)/r^2 + (m2*m3)/r^2 a2 = m1/r^2 + m3/r^2 |
(Equation 7) (Equation 8) |
The average distance from Pluto to the Sun is roughly 6*10^9 km. It takes roughly 2 hours for light to travel from that distance. That means on average we observe the planet Pluto 2 hours in the past.
When you want to do an accurate simulation this complexity has to be taken into account.
The basic idea behind Newton's Law is that the sum of all the forces between all objects in a closed system is zero. This implies the forces induced by all the objects in the entire universe. From a practical point this are all the stars in our Galaxy. All those forces for one object (our Sun) don't cancel. The resulting force to make Our Galaxy stable (in equilibrium) is the force that drives our Sun around the center of Our Galaxy.
However there is one more complication. Consider two stars of equal mass which move in exact circle around each other. Consider a third much smaller which moves in a large circle in the same plane around the center of gravity of the two large stars.
Figure 1 shows this situation, with m1=m2 and m3 is small
v1 .m1---> . . . . C . m3 . | . | v3 . V <--- m2. v2 Figure 3 - 3 objects |
The next question is what is the speed of gravity (of the gravitational field or of gravity waves). Accordingly to General Relativity the speed is equal to the speed of light.
IMO Special Relativity nor General Relativity make any claim what the value of the speed of light c is. This value should be calculated based on observations and is supposed to be constant. The name is also misleading. What we are discussing is in general the speed of an electro-magnetic field not the speed of light.
In the above examples we have studied the movement of two or three objects. In reality the objects could be planets, stars, galaxies but also black holes. All these objects move accordingly to Newton's Law if the objects are considered point masses.
What all these objects do they create a gravitational field. Such a field transmits information by means of hypothetical particles called gravitons. For Black holes this is important because as the name implies black holes can not physical be observed because they don't transmit photons (light) but they can be gravitational be detected. Within our Galaxy there is a Black hole because we can observe stars with rotate around the center of our Galaxy which is "invisible". Using those stars the mass of the Black hole can be calculated.
What that means is that gravitons behave completely different as photons, implying that the speed which these particles propagate (gravitons versus photons) can be different.
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