tdi_for_glitches.ipynb

tdi_for_glitches.ipynb

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        "<a href=\"https://colab.research.google.com/gist/avivajpeyi/3083790412049a4e33fff9d7e5ce6767/tdi_for_glitches.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "# TDI for glitches\n",
        "\n",
        "For each of the two optical benches per MOSA, there are associated intermediary variables $\\xi_{ij}$ and $\\eta_{ij}$ constructed to mitigate spacecraft jitter and to reduce the six laser links to three independent effective links.\n",
        "The $\\xi_{ij}$ combinations have the form\n",
        "\\begin{equation}\n",
        "    \\xi_{12}\n",
        "    =\n",
        "    \\text{isi}_{12}\n",
        "    +\n",
        "    \\frac{\\text{rfi}_{12}-\\text{tmi}_{12}}{2}\n",
        "    +\n",
        "    \\frac{\\textbf{D}(\\text{rfi}_{21}-\\text{tmi}_{21})}{2},\n",
        "\\end{equation}\n",
        "whose subsequent index combinations $(ij)$ are obtained by rotation and reflection of the indices.\n",
        "The $\\eta_{ij}$ combinations have the form\n",
        "\\begin{equation}\n",
        "    \\eta_{12}\n",
        "    =\n",
        "    \\xi_{12}\n",
        "    +\n",
        "    \\frac{\\textbf{D}(\\text{rfi}_{21}-\\text{rfi}_{23})}{2},\\\\\n",
        "    \\eta_{13}\n",
        "    =\n",
        "    \\xi_{13}\n",
        "    +\n",
        "    \\frac{\\text{rfi}_{12}-\\text{rfi}_{13}}{2},\n",
        "\\end{equation}\n",
        "whose subsequent index combinations are obtained by cyclic permutation.\n",
        "\n",
        "\n",
        "\n",
        "We can then express the first-generation Michelson variable for a static LISA configuration as\n",
        "\\begin{equation}\n",
        "    X_1\n",
        "    \\approx\n",
        "    (1-\\textbf{D}^2)[\\eta_{13}+\\textbf{D}\\eta_{31} - \\eta_{12} - \\textbf{D}\\eta_{21}],\n",
        "\\end{equation}\n",
        "and obtain $Y_1$ and $Z_1$ by cyclic permutation of indices.\n",
        "The second-generation Michelson variable is\n",
        "\\begin{equation}\n",
        "    X_2 \\approx (1 - \\textbf{D}^4) X_{1}\n",
        "\\end{equation}\n",
        "\n",
        "\n",
        "We can write the final expression for $X_2$ by plugging in $X_1$, and the various $\\eta_{ij}$:\n"
      ],
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    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "colab": {
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        },
        "id": "mTPT273wF1Y8",
        "outputId": "21b789fd-e9cf-4f5c-bb6f-8cc6b6ea5c6c"
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      "outputs": [
        {
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          "text": [
            "Latex output:\n",
            "\\begin{equation}\n",
            "X_{2} = \\frac{\\left(\\mathbf{D} - 1\\right)^{2} \\left(\\mathbf{D} + 1\\right)^{2} \\left(\\mathbf{D}^{2} + 1\\right) \\left(\\mathbf{D}^{2} \\text{tmi}_{12} - \\mathbf{D}^{2} \\text{tmi}_{13} + 2 \\mathbf{D} \\text{tmi}_{21} - 2 \\mathbf{D} \\text{tmi}_{31} + \\text{tmi}_{12} - \\text{tmi}_{13}\\right)}{2}\n",
            "\\end{equation}\n"
          ]
        }
      ],
      "source": [
        "import sympy as sp\n",
        "\n",
        "# 1. Define symbols\n",
        "# ------------------------------------------------------------------\n",
        "D = sp.Symbol(\"D\", commutative=True)   # static equal-arm delay operator\n",
        "half = sp.Rational(1, 2)\n",
        "\n",
        "# Measurements (tmi_ij)\n",
        "t12, t13, t21, t31 = sp.symbols(\"tmi_{12} tmi_{13} tmi_{21} tmi_{31}\")\n",
        "\n",
        "# η_ij variables\n",
        "eta12, eta13, eta21, eta31 = sp.symbols(\"eta_{12} eta_{13} eta_{21} eta_{31}\")\n",
        "\n",
        "# 2. Glitch-only substitution: η_ij = -(1/2) (tmi_ij + D * tmi_ji)\n",
        "# ------------------------------------------------------------------\n",
        "eta_subs = {\n",
        "    eta12: -half * (t12 + D * t21),\n",
        "    eta13: -half * (t13 + D * t31),\n",
        "    eta21: -half * (t21 + D * t12),\n",
        "    eta31: -half * (t31 + D * t13),\n",
        "}\n",
        "\n",
        "# 3. First- and second-generation Michelson combinations\n",
        "# ------------------------------------------------------------------\n",
        "X1_eta = (1 - D**2) * (eta13 + D*eta31 - eta12 - D*eta21)\n",
        "X2_eta = (1 - D**4) * X1_eta\n",
        "\n",
        "# Substitute η_ij → tmi_ij\n",
        "X2_tmi = sp.simplify(sp.expand(X2_eta.subs(eta_subs)))\n",
        "X2_tmi = sp.simplify(sp.factor(X2_tmi))\n",
        "\n",
        "\n",
        "# 4. Print results\n",
        "# ------------------------------------------------------------------\n",
        "\n",
        "# LaTeX-ready equation\n",
        "latex_expr = sp.latex(sp.Eq(sp.Symbol(\"X_2\"), X2_tmi))\n",
        "latex_expr = latex_expr.replace(\"D\", \"\\\\mathbf{D}\")\n",
        "latex_expr = latex_expr.replace(\"tmi\", \"\\\\text{tmi}\")\n",
        "print(\"Latex output:\\n\\\\begin{equation}\")\n",
        "print(str(latex_expr))\n",
        "print(\"\\\\end{equation}\")\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Expression obtained from Sympy:\n",
        "\n",
        "\\begin{equation}\n",
        "X_{2} = \\frac{\\left(\\mathbf{D} - 1\\right)^{2} \\left(\\mathbf{D} + 1\\right)^{2} \\left(\\mathbf{D}^{2} + 1\\right) \\left(\\mathbf{D}^{2} \\text{tmi}_{12} - \\mathbf{D}^{2} \\text{tmi}_{13} + 2 \\mathbf{D} \\text{tmi}_{21} - 2 \\mathbf{D} \\text{tmi}_{31} + \\text{tmi}_{12} - \\text{tmi}_{13}\\right)}{2}\n",
        "\\end{equation}\n",
        "\n",
        "\n",
        "Latex expression obtained by hand:\n",
        "\n",
        "\\begin{equation}\n",
        "X_2\n",
        "= \\frac{1}{2}(1 - \\textbf{D}^4)^2\\bigl(\\text{tmi}_{12} - \\text{tmi}_{13}\\bigr)\n",
        "  + \\textbf{D}(1 - \\textbf{D}^4)(1 - \\textbf{D}^2)\\bigl(\\text{tmi}_{21} - \\text{tmi}_{31}\\bigr).\n",
        "\\end{equation}\n",
        "\n",
        "\n",
        "We can now check if the two match:\n"
      ],
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    {
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      "source": [
        "# Analytic target form for validation\n",
        "# ------------------------------------------------------------------\n",
        "X2_target = half*(1 - D**4)**2 * (t12 - t13) + D*(1 - D**4)*(1 - D**2)*(t21 - t31)\n",
        "\n",
        "# Check equivalence\n",
        "print(\"Does sympy match the hand derived expression?: \", sp.simplify(X2_tmi - X2_target) == 0)\n"
      ],
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        "outputId": "958b38b1-1f95-48f0-fa02-0fa0af9924a1"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Does sympy match the hand derived expression?:  True\n"
          ]
        }
      ]
    }
  ]
}