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Problem

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problem description

# Problem setup: A distant energetic astrophysical object contains a single, spherical emission blob moving relativistically down its jet with bulk Doppler factor δ\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, LX,limL_{X,\mathrm{lim}}, appears in the observed 0.3–10 keV X-ray band. --- **Symbols and parameters** | Symbol | Description | | ------ | ----------- | | zz | Source redshift | | tvt_v | Observer-frame variability time-scale | | $R'_b$ | Co-moving radius of the blob | | EsE_s | Characteristic synchrotron-photon energy (observer frame) | | LsL_s | Isotropic-equivalent synchrotron luminosity at EsE_s | | β\beta | X-ray photon index (Fε∝ε−βF_\varepsilon \propto \varepsilon^{-\beta}) | | EpE_p | Proton energy satisfying the photopion threshold | | EpLEpE_p L_{E_p} | Proton power per logarithmic bin at EpE_p | | σ^pπ\hat{\sigma}_{p\pi} | Inelasticity-weighted photopion cross-section | | ϵ¯Δ\bar{\epsilon}_\Delta | Photon energy (in proton rest frame) at the Δ(1232)\Delta(1232)-resonance peak (∼0.3GeV\sim 0.3\ \mathrm{GeV}) | | Δ¯\bar{\Delta} | Mean fractional proton energy transferred to pions | | fxf_x | Fraction of cascade luminosity emerging in 0.3 - 10 keV X-ray band | | LX,limL_{X,\mathrm{lim}} | Observational upper limit on 0.3 – 10 keV luminosity | | mpm_p | Proton mass | | cc | Speed of light | | f(β)f(\beta) | Spectral function f(β)=21+β(516+g(β)/2)=21+β(516+1200⋅30β−1)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}) | --- **Relevant physical relations** * **Causality / light-crossing** $$R'_b \approx \frac{c\,t_v\,\delta}{1+z}.$$ * **Delta-resonance threshold** EpEs≈mpc2ϵ¯Δ2(1+z)2δ2.E_p E_s \approx \frac{m_p c^{2}\,\bar{\epsilon}_\Delta}{2(1+z)^{2}}\,\delta^{2}. * **Cascade luminosity constraint** Lcascade,X=fx(EpLEp)τpγ≤LX,lim.L_{\mathrm{cascade},X} = f_x\,(E_pL_{E_p})\,\tau_{p\gamma}\;\le\;L_{X,\mathrm{lim}}. # Main problem: Assuming that a fraction fxf_x of the bolometric cascade luminosity will emerge in the X-ray band, derive the minimum Doppler factor δmin2+2β\delta_{\min}^{2 + 2\beta} of the emission region using only the quantities: zz, cc, tvt_v, mpm_p, EsE_s, EpLEpE_pL_{E_p}, LsL_s, LX,limL_{X,\mathrm{lim}}, σ^pπ\hat{\sigma}_{p\pi}, β\beta, Δ¯\bar{\Delta}, ϵ¯Δ\bar{\epsilon}_\Delta, and fxf_x. Your final answer should be a closed-form symbolic expression; do not insert numerical values.
Plain-text mathematical notation (without MathML)
# Problem setup:

A distant energetic astrophysical object contains a single, spherical emission blob moving relativistically down its jet with bulk Doppler factor δ. 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,lim), appears in the observed 0.3–10 keV X-ray band.

---

**Symbols and parameters**

| Symbol | Description |
| ------ | ----------- |
| z | Source redshift |
| t_(v) | Observer-frame variability time-scale |
| $R'_b$ | Co-moving radius of the blob |
| E_(s) | Characteristic synchrotron-photon energy (observer frame) |
| L_(s) | Isotropic-equivalent synchrotron luminosity at E_(s) |
| β | X-ray photon index (F_(ε)∝ε^(−β)) |
| E_(p) | Proton energy satisfying the photopion threshold |
| E_(p)L_(E_(p)) | Proton power per logarithmic bin at E_(p) |
| (σ)^_(pπ) | Inelasticity-weighted photopion cross-section |
| (ϵ)¯_(Δ) | Photon energy (in proton rest frame) at the Δ(1232)-resonance peak (∼0.3 GeV) |
| (Δ)¯ | Mean fractional proton energy transferred to pions |
| f_(x) | Fraction of cascade luminosity emerging in 0.3 - 10 keV X-ray band |
| L_(X,lim) | Observational upper limit on 0.3 – 10 keV luminosity |
| m_(p) | Proton mass |
| c | Speed of light |
| f(β) | Spectral function f(β)=(2)/(1+β)((5)/(16)+g(β)/2)=(2)/(1+β)((5)/(16)+(1)/(200)⋅30^(β−1)) |

---

**Relevant physical relations**

* **Causality / light-crossing**

  $$R'_b \approx \frac{c\,t_v\,\delta}{1+z}.$$

* **Delta-resonance threshold**

  E_(p)E_(s)≈(m_(p)c² (ϵ)¯_(Δ))/(2(1+z)²) δ².

* **Cascade luminosity constraint**

  L_(cascade,X)=f_(x) (E_(p)L_(E_(p))) τ_(pγ) ≤ L_(X,lim).

# Main problem:

Assuming that a fraction f_(x) of the bolometric cascade luminosity will emerge in the X-ray band, derive the minimum Doppler factor δ_(min)^(2+2β) of the emission region using only the quantities:

z, c, t_(v), m_(p), E_(s), E_(p)L_(E_(p)), L_(s), L_(X,lim), (σ)^_(pπ), β, (Δ)¯, (ϵ)¯_(Δ), and f_(x).

Your final answer should be a closed-form symbolic expression; do not insert numerical values.
Original LaTeX notation
# Problem setup:

A 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.

---

**Symbols and parameters**

| Symbol | Description |
| ------ | ----------- |
| $z$ | Source redshift |
| $t_v$ | Observer-frame variability time-scale |
| $R'_b$ | Co-moving radius of the blob |
| $E_s$ | Characteristic synchrotron-photon energy (observer frame) |
| $L_s$ | Isotropic-equivalent synchrotron luminosity at $E_s$ |
| $\beta$ | X-ray photon index ($F_\varepsilon \propto \varepsilon^{-\beta}$) |
| $E_p$ | Proton energy satisfying the photopion threshold |
| $E_p L_{E_p}$ | Proton power per logarithmic bin at $E_p$ |
| $\hat{\sigma}_{p\pi}$ | Inelasticity-weighted photopion cross-section |
| $\bar{\epsilon}_\Delta$ | Photon energy (in proton rest frame) at the $\Delta(1232)$-resonance peak ($\sim 0.3\ \mathrm{GeV}$) |
| $\bar{\Delta}$ | Mean fractional proton energy transferred to pions |
| $f_x$ | Fraction of cascade luminosity emerging in 0.3 - 10 keV X-ray band |
| $L_{X,\mathrm{lim}}$ | Observational upper limit on 0.3 – 10 keV luminosity |
| $m_p$ | Proton mass |
| $c$ | Speed of light |
| $f(\beta)$ | Spectral function $f(\beta)=\frac{2}{1+\beta}(\frac{5}{16} + g(\beta)/2)=\frac{2}{1+\beta}(\frac{5}{16}+\frac{1}{200}\cdot 30^{\beta-1})$ |

---

**Relevant physical relations**

* **Causality / light-crossing**

  $$R'_b \approx \frac{c\,t_v\,\delta}{1+z}.$$

* **Delta-resonance threshold**

  $$E_p E_s \approx \frac{m_p c^{2}\,\bar{\epsilon}_\Delta}{2(1+z)^{2}}\,\delta^{2}.$$

* **Cascade luminosity constraint**

  $$L_{\mathrm{cascade},X} = f_x\,(E_pL_{E_p})\,\tau_{p\gamma}\;\le\;L_{X,\mathrm{lim}}.$$

# Main problem:

Assuming 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:

$z$, $c$, $t_v$, $m_p$, $E_s$, $E_pL_{E_p}$, $L_s$, $L_{X,\mathrm{lim}}$, $\hat{\sigma}_{p\pi}$, $\beta$, $\bar{\Delta}$, $\bar{\epsilon}_\Delta$, and $f_x$.

Your final answer should be a closed-form symbolic expression; do not insert numerical values.

code template

Code

import sympy as sp
from sympy.physics.units import speed_of_light as c

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 = sp.symbols(
    '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')

f_beta = 2 / (1 + beta) * (5 / 16 + 1 / 200 * 30 ** (beta-1))
E_p = m_p * c**2 * bar_epsilon_Delta / (2 * (1 + z)**2) * bar_Delta**2 / E_s

def 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):
    r"""
    Return the expression of $\delta_{\min}^{2 + 2\beta}$ in Sympy format.

    Inputs
    ----------
    z                  : sympy.Symbol, source redshift, $z$
    c                  : sympy.Symbol, speed of light, $c$
    t_v                : sympy.Symbol, variability time-scale, $t_v$
    m_p                : sympy.Symbol, proton mass, $m_p$
    E_s                : sympy.Symbol, characteristic synchrotron photon energy, $E_s$
    E_p                : sympy.Symbol, proton energy satisfying the photopion threshold, $E_p$
    E_pL_Ep            : sympy.Symbol, proton power per logarithmic bin at $E_p$, $E_p L_{E_p}$
    L_s                : sympy.Symbol, isotropic-equivalent synchrotron luminosity at $E_s$, $L_s$
    L_X_lim            : sympy.Symbol, upper limit on 0.3 – 10 keV luminosity, $L_{X,\mathrm{lim}}$
    sigma_hat_p_pi     : sympy.Symbol, inelasticity-weighted photopion cross-section, $\hat{\sigma}_{p\pi}$
    beta               : sympy.Symbol, X-ray photon index, $\beta$
    bar_Delta          : sympy.Symbol, mean fractional proton energy transferred to pions, $\bar{\Delta}$
    bar_epsilon_Delta  : sympy.Symbol, photon energy (in proton rest frame) at the $\Delta(1232)$-resonance peak, $\bar{\epsilon}_\Delta$
    f_x                : sympy.Symbol, fraction of cascade luminosity emerging in 0.3 – 10 keV luminosity
    f_beta             : sympy.Symbol, spectral function, $f(\beta)$

    Outputs
    ----------
    delta_min_pow      : sympy.Expr, minimum Doppler factor of the emission region, $\delta_{\min}^{2 + 2\beta}$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    delta_min_pow = ...  # SymPy expression involving the inputs above
    # ---------------------------------------------------------------

    return delta_min_pow

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Source and history

upstream_id / Original problem ID
Challenge_25_main
task_id
b118f446-7ff9-5b5b-93a4-929977b80db3
task_key
train--Challenge~5f25~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import