The time dilation experienced by an astronaut in orbit around Earth due to the effects of both general and special relativity can be calculated using the following formula:
Δt' = Δt * sqrt(1 - (v²/c²)) * sqrt(1 - (2GM/(rc²)))
Where: Δt' is the dilated time experienced by the astronaut Δt is the proper time measured on Earth v is the velocity of the astronaut relative to Earth c is the speed of light in a vacuum G is the gravitational constant M is the mass of Earth r is the distance between the center of Earth and the astronaut
To simplify the calculation, let's assume a circular orbit around Earth with a radius of approximately 6,378 kilometers (average Earth radius). The velocity of the astronaut can be calculated using the formula for the velocity of a circular orbit:
v = sqrt(GM/r)
The values for the constants are: G = 6.67430 × 10^(-11) m³/kg/s² M = 5.972 × 10^24 kg c = 299,792,458 m/s
Now, let's plug in the values and calculate the time dilation:
r = 6,378,000 meters v = sqrt((6.67430 × 10^(-11) * 5.972 × 10^24) / 6,378,000) ≈ 7,905.6 m/s
Δt' = Δt * sqrt(1 - (v²/c²)) * sqrt(1 - (2GM/(rc²))) = Δt * sqrt(1 - ((7,905.6)² / (299,792,458)²)) * sqrt(1 - (2 * 6.67430 × 10^(-11) * 5.972 × 10^24) / (6,378,000 * (299,792,458)²))
Calculating this expression precisely requires high precision arithmetic and is best suited for numerical computation. The time dilation experienced by the astronaut due to general and special relativistic effects in orbit around Earth would be a very small fraction of a second.