In this paper, we use the metric coefficients and the equation of motion in the 2nd post-Newtonian approximation in scalar-tensor theory including intermediate range gravity to derive the deflection of light and compare it with previous works. These results will be useful for precision astrometry missions like GAIA (Global Astrometric Interferometer of Astrophysics), SIMS (The Space Interferometry Mission) and LATOR (Laser Astrometric Test Of Relativity) which aim at astrometry with microarcsecond and nanoarcsecond accuracies and need 2nd post-Newtonian framework and ephemeris to determine the stellar and spacecraft positions.
The relativistic light deflection passing near the solar rim is 1.75 as (arcsec). The first post-Newtonian approximation is valid to 10 -6 and the second post-Newtonian is valid to 10 -12 of relativistic effects such as light deflection in the solar system. For astrometry mission to measure angles with accuracy in the nas to µas range, 2nd post-Newtonian approximation of relevant theories of gravity is required both for the angular measurement and for relativistic gravity test. The scalar-tensor theory is widely discussed and used in tests of relativistic gravity. In order to confront the predictions of scalar-tensor theory with experiment in the solar system, it is necessary to compute it's second post-Newtonian approximation and certain gravitational effects such as deflection of light, time delay of light and perihelion shift in this approximation. The second post-Newtonian contribution for light ray has been discussed for a long time by and [1], [2] and by others later. In this paper, we use the metric coefficients we obtained earlier ( [3]) to compute the deflection in the second post-Newtonian approximation considering the velocity of the observer (spacecraft).
The calculation of light deflection to 2PN approximation requires knowledge of terms in the metric to order (v/c) 4 . For the scalar-tensor theory, the metric coefficients are
in the global coordinates. Note that U is given by
and the parameters γ, β, ξ 1 and Λ are given in [3] by
The basic equations of light ray read
where k µ ≡ dx µ /dt. Consider a light signal emitted at ( x 0 , t 0 ) in an initial direction described by the unit vector n satisfying n • n = 1 and let its have the form
where x p (t)and x pp (t) are the first and second post-Newtonian correction respectively. For the rotating sun, we obtain the solution needed to the second-order approximation by the iterative method. Consider an observer (satellite, spacecraft) with the four-velocity u µ who receives the signals from two different sources. The angle α between the directions of two incoming photons is given by the following expression:
where P µν = g µν + u µ u ν is a projection operator. Defining the angle δα to be the deflected angle from the original angle α 0 , and expanding cos α around α 0 to the second order, we have
where δα p and δα pp are the deflection angles for the first and second post-Newtonian approximations. After a straightforward but lengthy calculation, we have
for light passing the solar limb in the equatorial plane from outside the solar system. Here M is the solar mass, R the solar radius, J the solar angular momentum and J 2 the solar quadrupole moment parameter. The second term comes from the intermediate-range force and other terms agree with the former works. The 2PN light trajectory obtained here is useful for obtaining 2PN range of deep space laser ranging missions ASTROD I and ASTROD ( [4]).
A long paper on this topic will be presented in the future.
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