= Solution
In the weak-field <Zeeman effect>, <spin-orbit coupling> keeps $L,S,J$ good and a level shifts by $\mu_Bg_JBm_J$. At fields large compared with fine-structure coupling but still within the linear electronic Zeeman approximation, $L$ and $S$ decouple: the <Paschen-Back effect> gives shifts $\mu_BB(m_L+2m_S)$. Electric-dipole transitions obey $\Delta m_L=0,\pm1$, $\Delta m_S=0$ in that limit, recovering an orbital triplet pattern. Intermediate fields mix the coupled states.
For the weak-field <sodium D-line Zeeman pattern>, the given <Landé g-factor> formula gives
$$
g({}^2S_{1/2})=2,\qquad g({}^2P_{1/2})=2/3,\qquad g({}^2P_{3/2})=4/3.
$$
Allowed <photon> shifts are $\delta E=\mu_BB(g_um_u-2m_l)$ with $m_u-m_l=0,\pm1$. Thus
$$
\boxed{D_1:\ \delta E/(\mu_BB)=\pm2/3,\pm4/3,\qquad
D_2:\ \delta E/(\mu_BB)=\pm1/3,\pm1,\pm5/3.}
$$
These are the four- and six-component patterns. Both $\pi$ and $\sigma$ components are represented, so an observation strictly along the field, which removes $\pi$ emission, would not show this complete pattern. <Hyperfine structure> is neglected.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-gp2012-zeeman.png]
{title=Calculated sodium Zeeman shifts}
{description=Original stick positions calculated from selection rules and Landé factors, in units of magnetic energy. Equal plotted stick heights are illustrative and do not assert equal transition intensities. No source spectral figure is copied.}
To determine the <magnetic field from a Zeeman diagram>, the unsplit separation provides its <energy> scale:
$$
\Delta E_D=hc(1/\lambda_2-1/\lambda_1)=2.1315\times10^{-3}\,\mathrm{eV}.
$$
The smallest adjacent split spacing $s_E=2\mu_BB_1/3$ is approximately $0.070$ of that separation in the sketch. Therefore
$$
\boxed{B_1\simeq\frac{3(0.070)\Delta E_D}{2\mu_B}\simeq3.9\,\mathrm T\approx4\,\mathrm T.}
$$
This is a graphical estimate; the drawing does not warrant high precision. Its splitting <energy> is small compared with $\Delta E_D$, consistent with the weak-field approximation.
For <Doppler resolution of a Zeeman pattern>, thermal <velocities> give <Gaussian> <Doppler broadening> with width
$$
\frac{\Delta\nu_D}{\nu}=\sqrt{\frac{8k_BT\ln2}{m_{\rm Na}c^2}},\qquad m_{\rm Na}=\frac{0.0230}{N_A}\,\mathrm{kg}.
$$
At $350\,\mathrm K$, the photon-energy FWHM near $589\,\mathrm{nm}$ is $5.88\times10^{-6}\,\mathrm{eV}$. Equating the smallest adjacent Zeeman spacing to this width gives \b[$B\sim0.15\,\mathrm T$] for loss of individually resolved components. There is no sharp <Gaussian> resolution threshold; using whole-pattern width instead gives a smaller field of order $0.03$–$0.04\,\mathrm T$ for complete disappearance into a single envelope. Instrumental and collisional broadening would also affect the estimate.
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