CritPt / Challenge_25_main / A distant energetic astrophysical object contains a single, spherical emission…

# 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.

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**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)) |

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**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.

# 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 $f(\beta)=\frac{2}{1+\beta}(\frac{5}{16} + g(\beta)/2)=\frac{2}{1+\beta}(\frac{5}{16}+\frac{1}{200}\cdot 30^{\beta-1})$ |

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**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.