= Solution
<Inverse Compton scattering> transfers <energy> from fast <electrons> to lower-<energy> <photons>. In the <Thomson scattering> limit the scattering is approximately elastic in the <electron>'s instantaneous rest frame, but the transformations into and out of that frame can greatly increase the <photon> <energy> in the original frame.
In a <blazar>, relativistic jet <electrons> can scatter their own <synchrotron radiation>, producing <synchrotron self-Compton radiation>, or scatter <photons> from an <accretion disk> and its surrounding reprocessing material. The two spectral humps constrain <electron> <Lorentz factors>, the <electron> distribution and, with assumptions about geometry and relativistic beaming, the <magnetic field>. For the same <electron> population in the Thomson regime, the ratio of inverse-Compton to synchrotron power scales as $u_{\rm ph}/u_B$ in the plasma rest frame. It does not determine the field without information about the <photon energy density> and beaming.
In a <galaxy cluster>, hot thermal <electrons> scatter <cosmic microwave background> <photons> and slightly raise their mean <energy>, producing the <thermal Sunyaev-Zeldovich effect>. The spectral distortion measures the line-of-sight integral of <electron> <pressure>, approximately $y=\sigma_T\int n_ek_BT_e\,d\ell/(m_ec^2)$ for nonrelativistic thermal <electrons>. Combined with X-ray emission, which weights density approximately quadratically, it constrains the gas density, <temperature> and <mass> distribution. These are distinct source populations and seed <photon> fields from the blazar example.
For the power derivation, use an isotropic incident radiation field of \b[total <photon energy density>] $u_{\rm ph}$ in the laboratory frame. Its <stress-energy tensor> has $T^{00}=u_{\rm ph}$ and $T^{zz}=u_{\rm ph}/3$, with zero mean flux. Boosting to an <electron> moving at $\beta c$ along $z$ gives
$$
T'^{0z}=-\gamma^2\beta(T^{00}+T^{zz})=-\frac43\gamma^2\beta u_{\rm ph}.
$$
The radiation <energy flux> in this rest frame is $S'_z=cT'^{0z}$. Elastic <Thomson scattering> exerts the drag $F'_z=\sigma_TS'_z/c$. The parallel three-force has the same value in the laboratory frame, so the <electron>'s positive <energy> loss rate is
$$
\boxed{P_{\rm IC}=-F_z\beta c=\frac43\sigma_Tc\,u_{\rm ph}\beta^2\gamma^2.}
$$
This <isotropic inverse-Compton drag> assumes <photon> <energies> small compared with $m_ec^2$ in the <electron> rest frame, negligible recoil and an isotropic seed field in the specified frame. An anisotropic <photon> field requires its directional moments; the recoil-dependent <differential scattering cross-section> of <Compton scattering> becomes necessary at larger rest-frame <energies>.
There are two conventions to qualify in the printed expression. First, the density multiplying $4/3$ must be $u_{\rm ph}$, not literally only the magnetic part of a propagating electromagnetic field. That field has equal electric and magnetic <energy> densities, so $u_{\rm ph}=2u_{\rm mag}$ and the corresponding formula is $P_{\rm IC}=(8/3)\sigma_Tcu_{\rm mag}\beta^2\gamma^2$. Second, this is net <electron> <energy> loss, rather than all incident plus scattered <photon> <energy>. At $\beta=0$ an <electron> still scatters power $\sigma_Tcu_{\rm ph}$, but transfers no kinetic <energy> to the <photons>, so the inverse-Compton gain is zero.
For an individual incident <photon> of <energy> $\varepsilon$, the <electron>-frame <energy> is $\varepsilon'=\gamma\varepsilon(1-\beta\cos\theta)$. For a head-on <photon> followed by forward emission in the laboratory,
$$
\varepsilon_{\rm out}\simeq\gamma^2(1+\beta)^2\varepsilon\simeq4\gamma^2\varepsilon.
$$
Averaging an isotropic seed field gives $\langle\varepsilon_{\rm out}\rangle\simeq(4/3)\gamma^2\varepsilon$ for ultrarelativistic <electrons>. The original plot has an inverse-Compton maximum of order $10^8\,\mathrm{eV}$, a synchrotron maximum around $10^{-1}\,\mathrm{eV}$ and a disk component around $1\,\mathrm{eV}$. Thus
$$
\boxed{\gamma\sim\sqrt{\frac{3\varepsilon_{\rm out}}{4\varepsilon_{\rm seed}}}\sim9\times10^3\ \hbox{to}\ 3\times10^4.}
$$
The head-on maximum gives the same order of magnitude, about $5\times10^3$ to $1.6\times10^4$. These are approximate readings of a broad spectrum, not precise fitted parameters. Disk and synchrotron seed <photons> need not be scattered by the same part of the <electron> distribution, and bulk jet Doppler factors also enter the observed peak relation. The indicated rest-frame seed <energies> remain well below the <electron> rest <energy> in this estimate.
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