(app:b)=
# Common Units and Numerical Scales
These tables collect the units and orders of magnitude most often needed for writing, estimates, and review. For the actual formulas, return to the first definition in the main text; when estimating, check units, scale, and validity range together.

(appsec:b-constants)=
## Basic Constants and Astronomical Conversions
| Quantity | Common value | Reminder |
|:---------------|:---------------------|:--------------------------|
| Speed of light | $c=2.998\times10^8\,{\rm m\,s^{-1}}$ | Baseline light-travel time $B/c$: $1\,{\rm km}$ is about $3.3\,\mu{\rm s}$, and $1000\,{\rm km}$ is about $3.3\,{\rm ms}$. |
| Planck constant | $h=6.626\times10^{-34}\,{\rm J\,s}$ | Single-photon energy is $E=h\nu=hc/\lambda$. |
| Boltzmann constant | $k_{\rm B}=1.381\times10^{-23}\,{\rm J\,K^{-1}}$ | Needed for brightness temperature, Planck occupation number, and thermal noise. |
| Jansky | $1\,{\rm Jy}=10^{-26}\,{\rm W\,m^{-2}\,Hz^{-1}}$ | AB magnitudes and photon-rate estimates often start from Jy. |
| Parsec | $1\,{\rm pc}=3.086\times10^{16}\,{\rm m}$ | $1\,{\rm AU}$ subtends $1''$ at $1\,{\rm pc}$. |
| Solar radius | $R_\odot=6.96\times10^8\,{\rm m}$ | Useful for stellar angular diameters and binary scales. |
| Solar luminosity | $L_\odot=3.83\times10^{26}\,{\rm W}$ | Often combined with $F_{\rm bol}$, $T_{\rm eff}$, and $\theta$. |

(appsec:b-angle)=
## Angles, Wavelengths, and Baselines
| Quantity | Common value | Main-text location |
|:---------------|:---------------------|:--------------------------|
| Angle conversions | $1\,{\rm rad}=206265''$; $1\,{\rm mas}=4.848\times10^{-9}\,{\rm rad}$; $1\,\mu{\rm as}=4.848\times10^{-12}\,{\rm rad}$ | Chapters {ref}`chap:01` and {ref}`chap:05`. |
| Visible wavelengths | Blue-light intensity interferometry often uses $400$--$450\,{\rm nm}$; broad-band imaging often uses $500$--$800\,{\rm nm}$ | Chapters {ref}`chap:05` and {ref}`chap:18`. |
| Diffraction scale | $\lambda/B$ is about $1\,{\rm mas}$ at $500\,{\rm nm}$ and $100\,{\rm m}$ | Hundred-meter arrays are well matched to bright-star angular diameters. |
| First-null scale | A $1\,{\rm mas}$ uniform disk has its first null near $100\,{\rm m}$ at $416\,{\rm nm}$ | Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-first-null`. |
| $\mu{\rm as}$ structure | $10\,\mu{\rm as}$ at $500\,{\rm nm}$ corresponds to $\lambda/\theta\sim10\,{\rm km}$ | Nearby supernovae, compact AGN structure, and future long-baseline cases. |
| Projected baseline | $B_\perp$ changes with target elevation, hour angle, and array geometry | A fixed physical baseline is not a substitute for the full overnight $u,v$ coverage. |

(appsec:b-time)=
## Time, Frequency, and Coherence
| Quantity | Common value | Main-text location |
|:---------------|:---------------------|:--------------------------|
| Frequency conversion | $500\,{\rm nm}$ corresponds to $6.0\times10^{14}\,{\rm Hz}$ | Optical bandwidths often require conversion between $\Delta\lambda$ and $\Delta\nu$. |
| Optical coherence time | A $10\,{\rm nm}$-class filter in the visible typically gives $10^{-14}$--$10^{-13}\,{\rm s}$ | Chapter {ref}`chap:01`, Eq. {eq}`eq:coherence-time`. |
| Correlation bin | Tabletop experiments can use ps--ns bins; astronomical intensity interferometry is often limited by electronic response and data rate | Chapters {ref}`chap:07` and {ref}`chap:20`. |
| Detector jitter | Good SPAD, PMT, and TDC systems can reach tens of ps to ns; system synchronization must be calibrated separately | Chapter {ref}`chap:06`. |
| Dead time | SPAD dead time is often ns--$\mu{\rm s}$; PMTs and electronics also have recovery times | Dead time creates negative correlations at short delays. |
| Afterpulsing | Probabilities can be $10^{-4}$--$10^{-2}$ | Produces positive correlations at detector-specific delays. |
| Integration-time scaling | Pure statistical errors usually fall as $T^{-1/2}$ | Once a systematic floor is reached, longer integration does not automatically improve the result. |

(appsec:b-photon-rate)=
## Photon Rates, Magnitudes, and Backgrounds
| Quantity | Common value | Main-text location |
|:---------------|:---------------------|:--------------------------|
| AB-magnitude flux | AB zero point and photon-rate estimates are given in Chapter {ref}`chap:01`, Eqs. {eq}`eq:ab-rate` and {eq}`eq:photon-rate` | A proposal should state bandwidth, efficiency, collecting area, and background together. |
| Magnitude change | If a target is fainter by $1\,{\rm mag}$, the photon rate falls by about $10^{-0.4}\simeq0.40$ | Chapters {ref}`chap:18` and {ref}`chap:22`. |
| Telescope area | An ideal $D=12\,{\rm m}$ aperture has geometric area about $113\,{\rm m^2}$; the effective area must also include throughput and obscuration | Cherenkov-telescope optics, filters, and detector QE all enter. |
| Background dilution | If the target flux fraction is $f$, the second-order correlation excess is approximately suppressed by $f^2$ | Chapter {ref}`chap:01`, Eq. {eq}`eq:background-dilution`. |
| Narrow-line observations | A narrower line gives a longer coherence time, but total photon number and filter leakage may worsen | Be-star disks, maser/laser candidates, and BLR cases should all report line/continuum separation. |
| Night sky | Varies with Moon phase, zenith angle, filter, and field stop | A dark-field background is not valid for every target position. |

(appsec:b-instrument)=
## Instruments and Data Volume
| Quantity | Common value | Main-text location |
|:---------------|:---------------------|:--------------------------|
| Single-stream sampled data rate | $\Delta t=4\,{\rm ns}$, $16$ bits, and one channel give about $4\,{\rm Gb\,s^{-1}}$ | Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-data-rate`. |
| Multichannel expansion | 64 spectral channels push the same example to about $256\,{\rm Gb\,s^{-1}}$ per telescope | Requires real-time correlation, compression, or on-chip accumulation. |
| Number of baselines | 4 telescopes give 6 baselines; 60 telescopes give 1770 baselines | Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-baseline-number`. |
| Calibrator stars | Should be close to the target in sky position, color, and brightness, with known angular diameter | Chapter {ref}`chap:18`. |
| Quality slicing | Split by transparency, target elevation, background rate, dead time, and channel state | Reporting wall-clock time alone is not enough to reproduce an experiment. |
| Systematic floor | A correlation-amplitude floor of $10^{-5}$--$10^{-4}$ can dominate many SII targets | Chapter {ref}`chap:21`. |

(appsec:b-astro)=
## Astrophysical Orders of Magnitude
| Quantity | Common value | Main-text location |
|:---------------|:---------------------|:--------------------------|
| Bright-star angular diameters | Nearby bright stars are often $0.2$--$5\,{\rm mas}$ | Chapters {ref}`chap:10` and {ref}`chap:19`. |
| Early Type Ia velocities | Photospheric velocities are often $8000$--$15000\,{\rm km\,s^{-1}}$ | Chapters {ref}`chap:13` and {ref}`chap:20`. |
| Nearby supernova angular radius | At $20\,{\rm Mpc}$, $v=10^4\,{\rm km\,s^{-1}}$, and 15 days, the angular radius is a few $\mu{\rm as}$ | Kilometer to ten-kilometer baselines are needed for substantial information. |
| Crab period | About $33\,{\rm ms}$ | Phase-resolved photon statistics require absolute timing. |
| AGN broad-line region | Reverberation delays are often days to months, while angular scales are usually very small | Combine reverberation, angular displacement, and model priors. |
| CMB temperature | $2.725\,{\rm K}$ | Chapter {ref}`chap:16`; optical photon statistics cannot be transferred directly. |