kernel_dynamical_time

formal source

 392self-similarity exponent `α`. For a stationary orbit `T_dyn` is constant,
 393so the enhancement is constant, and the acceleration on an isolated mass
 394does not grow in time. -/
 395noncomputable def kernel_dynamical_time (P : KernelParams) (T_dyn : ℝ) : ℝ :=
 396  1 + P.C * (max 0.01 (T_dyn / P.tau0)) ^ P.alpha
 397
 398/-- The dynamical-time kernel is positive. -/
 399theorem kernel_dynamical_time_pos (P : KernelParams) (T_dyn : ℝ) :
 400    0 < kernel_dynamical_time P T_dyn := by
 401  unfold kernel_dynamical_time
 402  have hmax_pos : 0 < max 0.01 (T_dyn / P.tau0) := by
 403    apply lt_max_of_lt_left; norm_num
 404  have hpow_nonneg : 0 ≤ (max 0.01 (T_dyn / P.tau0)) ^ P.alpha :=
 405    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
 406  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (T_dyn / P.tau0)) ^ P.alpha :=
 407    mul_nonneg P.C_nonneg hpow_nonneg
 408  linarith
 409
 410/-- The dynamical-time kernel is at least 1. -/
 411theorem kernel_dynamical_time_ge_one (P : KernelParams) (T_dyn : ℝ) :
 412    1 ≤ kernel_dynamical_time P T_dyn := by
 413  unfold kernel_dynamical_time
 414  have hmax_pos : 0 < max 0.01 (T_dyn / P.tau0) := by
 415    apply lt_max_of_lt_left; norm_num
 416  have hpow_nonneg : 0 ≤ (max 0.01 (T_dyn / P.tau0)) ^ P.alpha :=
 417    Real.rpow_nonneg (le_of_lt hmax_pos) P.alpha
 418  have hcorr_nonneg : 0 ≤ P.C * (max 0.01 (T_dyn / P.tau0)) ^ P.alpha :=
 419    mul_nonneg P.C_nonneg hpow_nonneg
 420  linarith
 421
 422/-- **No cumulative-time growth.** For a stationary orbit `T_dyn(t) = T_dyn`
 423constant in time `t`, the dynamical-time kernel is constant in time. This
 424resolves Beltracchi's concern (1): the gravitational acceleration on a
 425test particle in a stable orbit does not grow as `t^α`. -/