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So I was reading a post of "What would happen if the sun disappeared". An excerpt:

After eight minutes, the Earth would not only lose the sun's light and heat, but also its gravitational influence. No longer in solar orbit, the Earth would continue in a straight line into space at 67,062 miles per hour.

I can liken this to spinning a ball on a string, and if I release the string, then instantaneously the ball would continue in a straight line along its tangential trajectory. How long after the sun disappearing would it take for this to happen to the Earth? Instantly? Or at the speed of light?

While the Sun suddenly disappearing is certainly not likely, the central mass of an orbit could suddenly change in shape or magnitude due to say a large mass ejection. This can certainly be mathematically modeled.

In this case, would the orbit of the Earth continue undisturbed until information of the change reached the Earth? If so, how long would the delay in response be?

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    $\begingroup$ Gravity propagates at the speed of light, you can look at it like a wave. Your assumption that if you "cut the string" the ball would immediately fly off is wrong. It takes time for the system to "react", usually the lack of counter force would propagate at the speed of sound in the material. $\endgroup$ – Dragongeek Apr 13 '18 at 22:19
  • $\begingroup$ @Christoph I agree with you, but the StackExchange intentionally makes the question migrations so hard as possible with various rule. They have no real reason behind that. The rule what avoids the migration of this question is that migrations between betas are essentially forbidden (they have another reasoning behind this rule). Furthermore, having some redundancy in the system (i.e. topicality overlapping) serves also as a help to motivate the sites for a cooperative treatment of incomers. On this reason, I am typically for pro-migration decisions, but against the topic narrowing decisions. $\endgroup$ – peterh - Reinstate Monica Apr 13 '18 at 22:23
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    $\begingroup$ A previous question about retarded gravitational potential Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch? was very well received and also received an excellent answer by @DavidHammen. This question has also received one good answer so far. Any precise orbit calculation requires attention to principles of general relativity, it's one of the critical ingredients to JPL's Horizons for example. I've slightly modified the question and voted to reopen. $\endgroup$ – uhoh Apr 15 '18 at 5:10
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The current best theory describing the gravitation is Einstein's General Relativity.

It is a little bit more complex, as Newton's classical theory of the gravitation. In Newton's theory, the gravitation is instant.

In Einstein's theory, it propagates with $c$. $c$ is the speed of the light. It is not the result of anything, it is simply postulated so by the theory.

Since then, many sophisticated experiment proved the GR. The latest one, the detection of the gravitational waves, required complex computing to predict the exact signals, using the GR, and it passed. But there are also many others, too.

There is currently no accepted theory in physics with instant effects. In Newton's era, the worst problem of the classical gravity was that it assumed instantenous effect (and without a propagating medium).

In most orbit calculations, the classical gravity is far more than enough in precision. There are more significant effects (for example, the effect of third bodies) which are also often ignored. GR is needed only rarely, typically where the speed of the bodies nears the speed of light, and their mass nears a black hole.

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