cesiumfrog said:
It hadn't occurred to me that, by "weight", the OP meant anything other than what is read when standing on a scale.
The common definition of weight (in physics) is mass times gravitational acceleration. To most people, weight is what a scale measures, which is something quite different than this standard definition of weight. (Legally, weight is a synonym for mass, but that's a different issue).
So what do scales measure?
- Newtonian POV: Scales measure the net non-gravitational force acting on a body. The gravitational component of the net force is not measurable because there is no way to shield the gravitational force.
- General relativistic POV: Scales measure the net force acting on a body. There is no gravitational component of the net force as gravitation is a pseudo-force in general relativity.
The total apparent force acting on an object in some arbitrary (Newtonian) reference frame is
Ftot=Ff+S+W where
Ff is the total fictitious force in this frame, S is the object's scale weight (aka apparent weight), and
W is the object's true weight (mass times gravitational acceleration). Suppose the object is stationary in some reference frame. The total apparent force is necessarily zero in this frame. The object's scale weight (a vector) is thus
S=-(Ff+W), where
Ff is the fictitious force acting on the object in a reference frame in which the object is stationary. Any other means of computing the scale weight must necessarily yield the same result as that obtained by computing forces in a frame in which the object is stationary.
The topic of discussion is the weight (scale weight) of someone standing still on the surface of the Earth. This person is stationary in the acceleration and rotating Earth-fixed frame (e.g., a frame with origin at the center of the Earth and rotating with the Earth). As the desired quantity is the Sun's contribution to scale weight, I'll ignore rotational effects. Both the gravitational acceleration of a person and the Earth toward the Sun contributes to the person's scale weight. The net result:
<br />
\boldsymbol W_s=<br />
GM_sm_p<br />
\left(<br />
\frac{\boldsymbol r_s - \boldsymbol r_p}{||\boldsymbol r_s - \boldsymbol r_p||^3} -<br />
\frac{\boldsymbol r_s}{||\boldsymbol r_s||^3}<br />
\right)<br />
where \boldsymbol W_s is the Sun's contribution to a person's scale weight, GM_s is the Sun's standard gravitational parameter, m_p is the person's mass, \boldsymbol r_s is the vector from the center of the Earth to the center of the Sun, and \boldsymbol r_p is the vector from the center of the Earth to the person.
Let theta be the angle between \boldsymbol r_s and \boldsymbol r_p. Denote two orthogonal unit vectors, \hat x and \hat y such that \boldsymbol r_s = r_s \hat x and \boldsymbol r_p = r_e (\cos\theta \hat x+\sin\theta \hat y). Here
r_s (scalar) is the distance between the Earth and Sun, nominally 1AU and
r_e is the radius of the Earth (I'm assuming a spherical Earth for simplicity). After a bit of grinding, the Sun's contribution to a person's scale weight is, to first order,
<br />
\boldsymbol W_s\approx<br />
m_p\,\frac{GM_s}{r_s^2}\,\frac{r_e}{r_s}(2\cos\theta \hat x - \sin\theta \hat y)<br />
At theta=0 (Sun directly overhead), this is
2mpGMsre/rs^3 directed sunward (i.e., away from the center of the Earth: the person is a bit lighter). At theta=180
o (midnight at the equator on one of the equinoxes), this first order approximation has the same magnitude as the theta=0 value, but is directed anti-sunward. This is also directed away from the center of the Earth, once again making the person a bit lighter. The magnitude of the Sun's contribution is at a minimum at theta=90
o (sunrise and sunset). However, the direction at this minimum magnitude is inward: The person is a bit heavier.
Chronos said:
That seems a bit high to me. Clocks suggest a smaller difference.
Gravitational time dilation due to the Sun's gravity is fairly small because the distance between the Earth and Sun is much, much greater than the Sun's Schwarzschild radius (~3 km). The difference in gravitational time dilation at sunrise and noon is extremely small because this involves the difference between two very small and very similar numbers.
