{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"b118f446-7ff9-5b5b-93a4-929977b80db3","task_key":"train--Challenge~5f25~5fmain","task_revision_id":"1","upstream_id":"Challenge_25_main","short_description":"A distant energetic astrophysical object contains a single, spherical emission…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\nfrom sympy.physics.units import speed_of_light as c\\n\\nz, t_v, m_p, E_s, E_pL_Ep, L_s, L_X_lim, sigma_hat_p_pi, beta, bar_Delta, bar_epsilon_Delta, f_x = sp.symbols(\\n    'z, t_v, m_p, E_s, E_pL_Ep, L_s, L_X_lim, sigma_hat_p_pi, beta, bar_Delta, bar_epsilon_Delta, f_x')\\n\\nf_beta = 2 / (1 + beta) * (5 / 16 + 1 / 200 * 30 ** (beta-1))\\nE_p = m_p * c**2 * bar_epsilon_Delta / (2 * (1 + z)**2) * bar_Delta**2 / E_s\\n\\ndef answer(z, c, t_v, m_p, E_s, E_p, E_pL_Ep, L_s, L_X_lim, sigma_hat_p_pi, beta, bar_Delta, bar_epsilon_Delta, f_x, f_beta):\\n    r\\\"\\\"\\\"\\n    Return the expression of $\\\\delta_{\\\\min}^{2 + 2\\\\beta}$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    z                  : sympy.Symbol, source redshift, $z$\\n    c                  : sympy.Symbol, speed of light, $c$\\n    t_v                : sympy.Symbol, variability time-scale, $t_v$\\n    m_p                : sympy.Symbol, proton mass, $m_p$\\n    E_s                : sympy.Symbol, characteristic synchrotron photon energy, $E_s$\\n    E_p                : sympy.Symbol, proton energy satisfying the photopion threshold, $E_p$\\n    E_pL_Ep            : sympy.Symbol, proton power per logarithmic bin at $E_p$, $E_p L_{E_p}$\\n    L_s                : sympy.Symbol, isotropic-equivalent synchrotron luminosity at $E_s$, $L_s$\\n    L_X_lim            : sympy.Symbol, upper limit on 0.3 – 10 keV luminosity, $L_{X,\\\\mathrm{lim}}$\\n    sigma_hat_p_pi     : sympy.Symbol, inelasticity-weighted photopion cross-section, $\\\\hat{\\\\sigma}_{p\\\\pi}$\\n    beta               : sympy.Symbol, X-ray photon index, $\\\\beta$\\n    bar_Delta          : sympy.Symbol, mean fractional proton energy transferred to pions, $\\\\bar{\\\\Delta}$\\n    bar_epsilon_Delta  : sympy.Symbol, photon energy (in proton rest frame) at the $\\\\Delta(1232)$-resonance peak, $\\\\bar{\\\\epsilon}_\\\\Delta$\\n    f_x                : sympy.Symbol, fraction of cascade luminosity emerging in 0.3 – 10 keV luminosity\\n    f_beta             : sympy.Symbol, spectral function, $f(\\\\beta)$\\n\\n    Outputs\\n    ----------\\n    delta_min_pow      : sympy.Expr, minimum Doppler factor of the emission region, $\\\\delta_{\\\\min}^{2 + 2\\\\beta}$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    delta_min_pow = ...  # SymPy expression involving the inputs above\\n    # ---------------------------------------------------------------\\n\\n    return delta_min_pow\",\"problem_description\":\"# Problem setup:\\n\\nA distant energetic astrophysical object contains a single, spherical emission blob moving relativistically down its jet with bulk Doppler factor $\\\\delta$. Inside the blob, shock-accelerated protons interact with the blob's own synchrotron photon field. The resulting cascade emission converts part of the proton power into electromagnetic (EM) radiation. Only a fraction of that cascade luminosity, $L_{X,\\\\mathrm{lim}}$, appears in the observed 0.3–10 keV X-ray band.\\n\\n---\\n\\n**Symbols and parameters**\\n\\n| Symbol | Description |\\n| ------ | ----------- |\\n| $z$ | Source redshift |\\n| $t_v$ | Observer-frame variability time-scale |\\n| $R'_b$ | Co-moving radius of the blob |\\n| $E_s$ | Characteristic synchrotron-photon energy (observer frame) |\\n| $L_s$ | Isotropic-equivalent synchrotron luminosity at $E_s$ |\\n| $\\\\beta$ | X-ray photon index ($F_\\\\varepsilon \\\\propto \\\\varepsilon^{-\\\\beta}$) |\\n| $E_p$ | Proton energy satisfying the photopion threshold |\\n| $E_p L_{E_p}$ | Proton power per logarithmic bin at $E_p$ |\\n| $\\\\hat{\\\\sigma}_{p\\\\pi}$ | Inelasticity-weighted photopion cross-section |\\n| $\\\\bar{\\\\epsilon}_\\\\Delta$ | Photon energy (in proton rest frame) at the $\\\\Delta(1232)$-resonance peak ($\\\\sim 0.3\\\\ \\\\mathrm{GeV}$) |\\n| $\\\\bar{\\\\Delta}$ | Mean fractional proton energy transferred to pions |\\n| $f_x$ | Fraction of cascade luminosity emerging in 0.3 - 10 keV X-ray band |\\n| $L_{X,\\\\mathrm{lim}}$ | Observational upper limit on 0.3 – 10 keV luminosity |\\n| $m_p$ | Proton mass |\\n| $c$ | Speed of light |\\n| $f(\\\\beta)$ | Spectral function $f(\\\\beta)=\\\\frac{2}{1+\\\\beta}(\\\\frac{5}{16} + g(\\\\beta)/2)=\\\\frac{2}{1+\\\\beta}(\\\\frac{5}{16}+\\\\frac{1}{200}\\\\cdot 30^{\\\\beta-1})$ |\\n\\n---\\n\\n**Relevant physical relations**\\n\\n* **Causality / light-crossing**\\n\\n  $$R'_b \\\\approx \\\\frac{c\\\\,t_v\\\\,\\\\delta}{1+z}.$$\\n\\n* **Delta-resonance threshold**\\n\\n  $$E_p E_s \\\\approx \\\\frac{m_p c^{2}\\\\,\\\\bar{\\\\epsilon}_\\\\Delta}{2(1+z)^{2}}\\\\,\\\\delta^{2}.$$\\n\\n* **Cascade luminosity constraint**\\n\\n  $$L_{\\\\mathrm{cascade},X} = f_x\\\\,(E_pL_{E_p})\\\\,\\\\tau_{p\\\\gamma}\\\\;\\\\le\\\\;L_{X,\\\\mathrm{lim}}.$$\\n\\n# Main problem:\\n\\nAssuming that a fraction $f_x$ of the bolometric cascade luminosity will emerge in the X-ray band, derive the minimum Doppler factor $\\\\delta_{\\\\min}^{2 + 2\\\\beta}$ of the emission region using only the quantities:\\n\\n$z$, $c$, $t_v$, $m_p$, $E_s$, $E_pL_{E_p}$, $L_s$, $L_{X,\\\\mathrm{lim}}$, $\\\\hat{\\\\sigma}_{p\\\\pi}$, $\\\\beta$, $\\\\bar{\\\\Delta}$, $\\\\bar{\\\\epsilon}_\\\\Delta$, and $f_x$.\\n\\nYour final answer should be a closed-form symbolic expression; do not insert numerical values.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}