The gravitational dipole theory of Hadjukovic (2010) is based on the hypothesis that antimatter has a negative gravitational mass and thus falls upwards on Earth. Astrophysically, the model is similar to but more fundamental than Modified Newtonian Dynamics (MOND), with the Newtonian gravity $g_{_N}$ towards an isolated point mass boosted by the factor $\nu = 1 + \left( \alpha/x \right) \tanh \left( \sqrt{x}/\alpha \right)$, where $x \equiv g_{_N}/a_{_0}$ and $a_{_0} = 1.2 \times 10^{-10}$ m/s$^2$ is the MOND acceleration constant. We show that $\alpha$ must lie in the range ${0.4-1}$ to acceptably fit galaxy rotation curves. In the Solar System, this interpolating function implies an extra Sunwards acceleration of ${\alpha a_{_0}}$. This would cause Saturn to deviate from Newtonian expectations by ${7000 \left( \alpha/0.4 \right)}$ km over 15 years, starting from known initial position and velocity on a near-circular orbit. We demonstrate that this prediction should not be significantly altered by the postulated dipole haloes of other planets due to the rather small region in which each planet's gravity dominates over that of the Sun. The orbit of Saturn should similarly be little affected by a possible ninth planet in the outer Solar System and by the Galactic gravity causing a non-spherical distribution of gravitational dipoles several kAU from the Sun. Radio tracking of the Cassini spacecraft orbiting Saturn yields a ${5\sigma}$ upper limit of 160 metres on deviations from its conventionally calculated trajectory. These measurements imply a much more stringent upper limit on $\alpha$ than the minimum required for consistency with rotation curve data. Therefore, no value of $\alpha$ can simultaneously match all available constraints, falsifying the gravitational dipole theory in its current form at extremely high significance.

中文翻译：

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太阳系对引力偶极子的限制

Hadjukovic (2010) 的引力偶极子理论基于反物质具有负引力质量并因此向上落到地球上的假设。在天体物理学上，该模型与修正牛顿动力学 (MOND) 类似，但更基本，牛顿引力 $g_{_N}$ 朝向孤立点质量，由因子 $\nu = 1 + \left( \alpha/x \right) \tanh \left( \sqrt{x}/\alpha \right)$，其中 $x \equiv g_{_N}/a_{_0}$ 和 $a_{_0} = 1.2 \times 10^{- 10}$ m/s$^2$ 是 MOND 加速度常数。我们表明 $\alpha$ 必须在 ${0.4-1}$ 范围内才能可接受地拟合星系旋转曲线。在太阳系中，这个插值函数意味着 ${\alpha a_{_0}}$ 的额外向日加速度。这将导致土星偏离牛顿预期 ${7000 \left( \alpha/0. 4 \right)}$ km 超过 15 年，从近圆形轨道上的已知初始位置和速度开始。我们证明，由于每个行星的引力在太阳引力上占主导地位的区域相当小，因此其他行星的假设偶极晕不应显着改变这一预测。土星的轨道应该同样受到太阳系外可能存在的第九颗行星和银河引力的影响，导致引力偶极子的非球形分布距太阳几kAU。对围绕土星运行的卡西尼号航天器的无线电跟踪产生了 160 米的上限 ${5\sigma}$ 与其常规计算轨迹的偏差。这些测量意味着 $\alpha$ 的上限比与旋转曲线数据一致所需的最小值要严格得多。