{ "schema": "story-eq/1", "generator": "scripts/story_eq_build.mjs", "katex": "0.16.47", "css": "vendor/katex/katex.css", "note": "Rendered at build time with KaTeX (MIT); the runtime ships only vendor/katex/katex.css and the fonts listed here. Bound numbers are placeholders (0.00 x 10^0) rewritten by js/eq/bind.js from story data keys.", "fonts": [ "KaTeX_Caligraphic-Regular.woff2", "KaTeX_Main-Regular.woff2", "KaTeX_Math-Italic.woff2", "KaTeX_Size1-Regular.woff2", "KaTeX_Size3-Regular.woff2" ], "eqs": { "E1": { "id": "E1", "title": "Column density", "svg": null, "beats": [ "B2", "B7b" ], "terms": { "N": { "label": "column density", "unit": "per square centimetre, log", "bind": { "src": "cell", "field": "logN" }, "fmt": "pow2", "prop": "N" }, "n": { "label": "ion density", "unit": "per cubic centimetre", "bind": { "src": "derived", "fn": "n_ion" }, "fmt": "sci3", "prop": "n" }, "dl": { "label": "path length through the cell", "unit": "kiloparsecs", "bind": { "src": "cell", "field": "dl_kpc" }, "fmt": "sig3", "prop": "dl", "drag": { "scale": "log", "rel": [ 0.1, 10 ], "step": 0.05, "big": 0.5 } }, "nH": { "label": "hydrogen density", "unit": "per cubic centimetre", "bind": { "src": "cell", "field": "nH" }, "fmt": "sci3", "prop": "nH" }, "Z": { "label": "metallicity", "unit": "times solar", "bind": { "src": "cell", "field": "Z" }, "fmt": "sig3", "prop": "Z", "ions": [ "OVI" ], "drag": { "scale": "log", "abs": [ 0.01, 3 ], "step": 0.05, "big": 0.5 } }, "f": { "label": "ion fraction", "unit": "", "bind": { "src": "cell", "field": "f_ion" }, "fmt": "sig3", "prop": "f" } }, "variants": { "OVI": { "val": { "tex": "{@t{n}{n_{\\rm O\\,VI}} = @v{n}\\ \\mathrm{cm^{-3}}}\\ \\times\\ {@t{dl}{d\\ell} = @v{dl}\\ \\mathrm{kpc}}\\ \\ \\Rightarrow\\ \\ {@t{N}{N} = @v{N}\\ \\mathrm{cm^{-2}}}", "html": "nOVI​=0.00×100cm−3×dℓ=0.00×100kpc⇒N=0.00×100cm−2", "terms": [ "n", "dl", "N" ], "values": [ "n", "dl", "N" ], "box_em": { "w": 39.344, "h": 1.522 } }, "sym": { "tex": "{@t{N}{N} = \\sum_{\\rm cells} @t{n}{n_{\\rm ion}}\\,@t{dl}{d\\ell}},\\qquad {n(\\text{O\\,VI}) = @t{nH}{n_{\\rm H}}\\,(\\mathrm{O/H})_\\odot\\,@t{Z}{(Z/Z_\\odot)}\\,@t{f}{f_{\\rm O\\,VI}}(n_{\\rm H},T)}", "html": "N=∑cells​nion​dℓ,n(OVI)=nH​(O/H)⊙​(Z/Z⊙​)fOVI​(nH​,T)", "terms": [ "N", "n", "dl", "nH", "Z", "f" ], "values": [], "box_em": { "w": 34.212, "h": 1.522 } }, "short": { "tex": "{@t{N}{N} = @v{N}\\ \\mathrm{cm^{-2}}}", "html": "N=0.00×100cm−2", "terms": [ "N" ], "values": [ "N" ], "box_em": { "w": 11.109, "h": 1.522 } }, "spoken": "Column density N is the sum over cells of the ion density times the path length; the O VI density is the hydrogen density times the solar oxygen abundance times the metallicity times the O VI fraction. This cell: {{n}} ions per cubic centimetre over {{dl}} kiloparsecs gives N of ten to the {{N}} per square centimetre." }, "HI": { "val": { "tex": "{@t{n}{n_{\\rm H\\,I}} = @v{n}\\ \\mathrm{cm^{-3}}}\\ \\times\\ {@t{dl}{d\\ell} = @v{dl}\\ \\mathrm{kpc}}\\ \\ \\Rightarrow\\ \\ {@t{N}{N} = @v{N}\\ \\mathrm{cm^{-2}}}", "html": "nHI​=0.00×100cm−3×dℓ=0.00×100kpc⇒N=0.00×100cm−2", "terms": [ "n", "dl", "N" ], "values": [ "n", "dl", "N" ], "box_em": { "w": 38.684, "h": 1.522 } }, "sym": { "tex": "{@t{N}{N} = \\sum_{\\rm cells} @t{n}{n_{\\rm ion}}\\,@t{dl}{d\\ell}},\\qquad {n(\\text{H\\,I}) = @t{f}{x_{\\rm HI}}\\,@t{nH}{n_{\\rm H}}}", "html": "N=∑cells​nion​dℓ,n(HI)=xHI​nH​", "terms": [ "N", "n", "dl", "f", "nH" ], "values": [], "box_em": { "w": 20.216, "h": 1.522 } }, "short": { "tex": "{@t{N}{N} = @v{N}\\ \\mathrm{cm^{-2}}}", "html": "N=0.00×100cm−2", "terms": [ "N" ], "values": [ "N" ], "box_em": { "w": 11.109, "h": 1.522 } }, "spoken": "Column density N is the sum over cells of the ion density times the path length; the neutral hydrogen density is the neutral fraction times the hydrogen density. This cell: {{n}} atoms per cubic centimetre over {{dl}} kiloparsecs gives N of ten to the {{N}} per square centimetre." } } }, "E2": { "id": "E2", "title": "Thermal width", "svg": "bwidth", "beats": [ "B5", "B7b", "B7c" ], "terms": { "b": { "label": "Doppler width", "unit": "kilometres per second", "bind": { "src": "cell", "field": "b" }, "fmt": "sig3", "prop": "b" }, "T": { "label": "temperature", "unit": "kelvin", "bind": { "src": "cell", "field": "T" }, "fmt": "grp", "prop": "T", "drag": { "scale": "log", "abs": [ 10000, 5000000 ], "step": 0.025, "big": 0.25 } }, "m": { "label": "atom mass", "unit": "", "bind": { "src": "derived", "fn": "b_alt" }, "fmt": "sig3", "prop": "b", "toggle": true }, "b_alt": { "label": "width of the other atom", "unit": "kilometres per second", "bind": { "src": "derived", "fn": "b_alt" }, "fmt": "sig3", "value": true } }, "variants": { "HI": { "val": { "tex": "{@t{b}{b_{\\rm H}} = @v{b}\\ \\mathrm{km\\,s^{-1}}}\\ \\ \\leftarrow\\ \\ {@t{T}{T} = @v{T}\\ \\mathrm{K}}\\qquad\\allowbreak {@t{m}{b_{\\rm O}} = @v{b_alt}\\ \\mathrm{km\\,s^{-1}}}", "html": "bH​=0.00×100kms−1←T=0.00×100KbO​=0.00×100kms−1", "terms": [ "b", "T", "m" ], "values": [ "b", "T", "b_alt" ], "box_em": { "w": 38.808, "h": 1.522 } }, "sym": { "tex": "{@t{b}{b} = \\sqrt{\\dfrac{2k\\,@t{T}{T}}{@t{m}{m}}}}\\qquad {(b_{\\rm turb} = 0)}", "html": "b=m2kT​​(bturb​=0)", "terms": [ "b", "T", "m" ], "values": [], "box_em": { "w": 13.753, "h": 2.952 } }, "short": { "tex": "{@t{T}{T} = @v{T}\\ \\mathrm{K}}", "html": "T=0.00×100K", "terms": [ "T" ], "values": [ "T" ], "box_em": { "w": 9.124, "h": 1.522 } }, "shortc": { "tex": "{@t{b}{b_{\\rm H}} = @v{b}\\ \\mathrm{km\\,s^{-1}}}\\ \\leftarrow\\ {@t{T}{T} = @v{T}\\ \\mathrm{K}}", "html": "bH​=0.00×100kms−1←T=0.00×100K", "terms": [ "b", "T" ], "values": [ "b", "T" ], "box_em": { "w": 23.687, "h": 1.522 } }, "spoken": "Thermal width b equals the square root of 2 k T over the atom's mass. Hydrogen at {{T}} kelvin: b is {{b}} kilometres per second; oxygen at the same temperature: {{b_alt}} kilometres per second." }, "OVI": { "val": { "tex": "{@t{b}{b_{\\rm O}} = @v{b}\\ \\mathrm{km\\,s^{-1}}}\\ \\ \\leftarrow\\ \\ {@t{T}{T} = @v{T}\\ \\mathrm{K}}\\qquad\\allowbreak {@t{m}{b_{\\rm H}} = @v{b_alt}\\ \\mathrm{km\\,s^{-1}}}", "html": "bO​=0.00×100kms−1←T=0.00×100KbH​=0.00×100kms−1", "terms": [ "b", "T", "m" ], "values": [ "b", "T", "b_alt" ], "box_em": { "w": 38.808, "h": 1.522 } }, "sym": { "tex": "{@t{b}{b} = \\sqrt{\\dfrac{2k\\,@t{T}{T}}{@t{m}{m}}}}\\qquad {(b_{\\rm turb} = 0)}", "html": "b=m2kT​​(bturb​=0)", "terms": [ "b", "T", "m" ], "values": [], "box_em": { "w": 13.753, "h": 2.952 } }, "short": { "tex": "{@t{T}{T} = @v{T}\\ \\mathrm{K}}", "html": "T=0.00×100K", "terms": [ "T" ], "values": [ "T" ], "box_em": { "w": 9.124, "h": 1.522 } }, "shortc": { "tex": "{@t{b}{b_{\\rm O}} = @v{b}\\ \\mathrm{km\\,s^{-1}}}\\ \\leftarrow\\ {@t{T}{T} = @v{T}\\ \\mathrm{K}}", "html": "bO​=0.00×100kms−1←T=0.00×100K", "terms": [ "b", "T" ], "values": [ "b", "T" ], "box_em": { "w": 23.707, "h": 1.522 } }, "spoken": "Thermal width b equals the square root of 2 k T over the atom's mass. Oxygen at {{T}} kelvin: b is {{b}} kilometres per second; hydrogen at the same temperature: {{b_alt}} kilometres per second." } } }, "E3": { "id": "E3", "title": "Optical depth at line centre", "svg": "tau0", "beats": [ "B4", "B7b", "B10" ], "terms": { "tau0": { "label": "optical depth at line centre", "unit": "", "bind": { "src": "cell", "field": "tau0" }, "fmt": "sig3", "prop": "depth", "drag": { "scale": "log", "via": "N", "step": 0.05, "big": 0.5 } }, "N": { "label": "column density", "unit": "per square centimetre, log", "bind": { "src": "cell", "field": "logN" }, "fmt": "pow2", "prop": "N", "drag": { "scale": "log10value", "ion": { "HI": [ 12, 19 ], "OVI": [ 11, 16 ] }, "step": 0.05, "big": 0.5 } }, "f": { "label": "oscillator strength", "unit": "", "bind": { "src": "atom", "field": "f" }, "fmt": "sig3", "prop": "f" }, "lam": { "label": "rest wavelength", "unit": "angstrom", "bind": { "src": "atom", "field": "lam0_A" }, "fmt": "fix3", "prop": "lam" }, "b": { "label": "Doppler width", "unit": "kilometres per second", "bind": { "src": "cell", "field": "b" }, "fmt": "sig3", "prop": "b" } }, "variants": { "any": { "val": { "tex": "{@t{tau0}{\\tau_0} = @v{tau0}}\\ \\ \\leftarrow\\ \\ {@t{N}{N} = @v{N}\\ \\mathrm{cm^{-2}}},\\allowbreak\\ \\ {@t{b}{b} = @v{b}\\ \\mathrm{km\\,s^{-1}}}", "html": "τ0​=0.00×100←N=0.00×100cm−2,b=0.00×100kms−1", "terms": [ "tau0", "N", "b" ], "values": [ "tau0", "N", "b" ], "box_em": { "w": 34.848, "h": 1.522 } }, "sym": { "tex": "{@t{tau0}{\\tau_0} = 1.4975\\times10^{-15}\\,\\dfrac{@t{N}{N}\\ @t{f}{f}\\ @t{lam}{\\lambda_{[\\text{\\AA}]}}}{@t{b}{b_{[\\mathrm{km/s}]}}}}", "html": "τ0​=1.4975×10−15b[km/s]​Nfλ[A˚]​​", "terms": [ "tau0", "N", "f", "lam", "b" ], "values": [], "box_em": { "w": 15.177, "h": 3.168 } }, "short": { "tex": "{@t{N}{N} = @v{N}\\ \\mathrm{cm^{-2}}}", "html": "N=0.00×100cm−2", "terms": [ "N" ], "values": [ "N" ], "box_em": { "w": 11.109, "h": 1.522 } }, "shortc": { "tex": "{@t{tau0}{\\tau_0} = @v{tau0}}\\ \\ \\propto\\ \\ {@t{N}{N}\\,f\\lambda\\,/\\,@t{b}{b}}", "html": "τ0​=0.00×100∝Nfλ/b", "terms": [ "tau0", "N", "b" ], "values": [ "tau0" ], "box_em": { "w": 15.214, "h": 1.522 } }, "spoken": "Optical depth at line centre tau zero equals 1.4975 times ten to the minus 15, times column density N, oscillator strength f and wavelength in angstrom, divided by the width b in kilometres per second. Here tau zero is {{tau0}}, from N of ten to the {{N}} and b of {{b}} kilometres per second." } } }, "E3b": { "id": "E3b", "title": "Line profile", "svg": "profile", "beats": [ "B4", "B10" ], "terms": { "tau0": { "label": "optical depth at line centre", "unit": "", "bind": { "src": "cell", "field": "tau0" }, "fmt": "sig3", "prop": "depth" }, "V": { "label": "line-of-sight velocity", "unit": "kilometres per second", "bind": { "src": "cell", "field": "V" }, "fmt": "fix1s", "prop": "V", "drag": { "scale": "lin", "rel": [ -200, 200 ], "clamp": [ -400, 400 ], "step": 1, "big": 10 } }, "b": { "label": "Doppler width", "unit": "kilometres per second", "bind": { "src": "cell", "field": "b" }, "fmt": "sig3", "prop": "b" }, "a": { "label": "damping parameter", "unit": "", "bind": { "src": "cell", "field": "a" }, "fmt": "sci3", "prop": "wings" } }, "variants": { "any": { "val": { "tex": "{@t{V}{V} = @v{V}\\ \\mathrm{km\\,s^{-1}}},\\allowbreak\\ \\ {@t{b}{b} = @v{b}\\ \\mathrm{km\\,s^{-1}}},\\allowbreak\\ \\ {@t{a}{a} = @v{a}}", "html": "V=0.00×100kms−1,b=0.00×100kms−1,a=0.00×100", "terms": [ "V", "b", "a" ], "values": [ "V", "b", "a" ], "box_em": { "w": 33.113, "h": 1.522 } }, "sym": { "tex": "{\\tau(v) = @t{tau0}{\\tau_0}\\,H\\!\\left(@t{a}{a},\\,\\dfrac{v - @t{V}{V}}{@t{b}{b}}\\right)},\\qquad {@t{a}{a} = \\dfrac{\\Gamma\\lambda}{4\\pi b}}", "html": "τ(v)=τ0​H(a,bv−V​),a=4πbΓλ​", "terms": [ "tau0", "a", "V", "b" ], "values": [], "box_em": { "w": 20.093, "h": 2.904 } }, "short": { "tex": "{@t{V}{V} = @v{V}\\ \\mathrm{km\\,s^{-1}}}", "html": "V=0.00×100kms−1", "terms": [ "V" ], "values": [ "V" ], "box_em": { "w": 11.767, "h": 1.522 } }, "spoken": "The optical depth profile is tau zero times the Voigt function H of a and v minus V over b, with damping parameter a equal to Gamma lambda over 4 pi b. Here the dip sits at {{V}} kilometres per second with width {{b}} and damping parameter {{a}}." } } }, "E4": { "id": "E4", "title": "Doppler shift", "svg": "doppler", "beats": [ "B6" ], "terms": { "V": { "label": "velocity relative to the galaxy", "unit": "kilometres per second", "bind": { "src": "cell", "field": "V" }, "fmt": "fix1s", "prop": "V" }, "lam": { "label": "observed wavelength", "unit": "angstrom", "bind": { "src": "derived", "fn": "lam_obs" }, "fmt": "fix3", "prop": "V", "drag": { "scale": "lin", "via": "V", "rel": [ -200, 200 ], "clamp": [ -400, 400 ], "step": 1, "big": 10 } }, "lam0": { "label": "rest wavelength", "unit": "angstrom", "bind": { "src": "atom", "field": "lam0_A" }, "fmt": "fix3", "prop": "lam" }, "vgal": { "label": "galaxy velocity", "unit": "kilometres per second", "bind": { "src": "derived", "fn": "v_gal" }, "fmt": "fix1s", "prop": "V" } }, "variants": { "any": { "val": { "tex": "{@t{lam}{\\lambda} = @v{lam}\\ \\text{\\AA}}\\ \\ \\Rightarrow\\ \\ {@t{V}{V} = @v{V}\\ \\mathrm{km\\,s^{-1}}}\\qquad\\allowbreak {@t{vgal}{v_{\\rm gal}} = @v{vgal}\\ \\mathrm{km\\,s^{-1}}}", "html": "λ=0.00×100A˚⇒V=0.00×100kms−1vgal​=0.00×100kms−1", "terms": [ "lam", "V", "vgal" ], "values": [ "lam", "V", "vgal" ], "box_em": { "w": 38.879, "h": 1.567 } }, "sym": { "tex": "{@t{V}{V} = c\\left(\\dfrac{@t{lam}{\\lambda}}{@t{lam0}{\\lambda_0}} - 1\\right) - @t{vgal}{v_{\\rm gal}}}", "html": "V=c(λ0​λ​−1)−vgal​", "terms": [ "V", "lam", "lam0", "vgal" ], "values": [], "box_em": { "w": 11.899, "h": 2.904 } }, "short": { "tex": "{@t{V}{V} = @v{V}\\ \\mathrm{km\\,s^{-1}}}", "html": "V=0.00×100kms−1", "terms": [ "V" ], "values": [ "V" ], "box_em": { "w": 11.767, "h": 1.522 } }, "spoken": "Velocity V equals the speed of light times the observed over the rest wavelength minus one, minus the galaxy's own velocity. Observed at {{lam}} angstrom, the gas moves at {{V}} kilometres per second relative to the galaxy, whose velocity is {{vgal}} kilometres per second." } } }, "E5": { "id": "E5", "title": "Optical depths add", "svg": "tausum", "beats": [ "B7d", "B11" ], "terms": { "taui": { "label": "optical depth of each cell", "unit": "", "bind": null, "prop": "cells" }, "taup": { "label": "peak optical depth so far", "unit": "", "bind": { "src": "live", "field": "tau_peak", "fallback": { "src": "totals", "field": "peak_tau", "when": "totals" } }, "fmt": "sig3", "prop": "depth" }, "F": { "label": "transmitted fraction at the deepest point", "unit": "", "bind": { "src": "derived", "fn": "F_min" }, "fmt": "F", "prop": "depth" } }, "variants": { "any": { "val": { "tex": "{@t{taup}{\\tau_{\\rm peak}} = @v{taup}}\\ \\ \\Rightarrow\\ \\ {@t{F}{F_{\\rm min}} = @v{F}}", "html": "τpeak​=0.00×100⇒Fmin​=0.00×100", "terms": [ "taup", "F" ], "values": [ "taup", "F" ], "box_em": { "w": 21.748, "h": 1.522 } }, "sym": { "tex": "{\\tau_{\\rm total}(v) = \\sum_i @t{taui}{\\tau_i(v)}},\\qquad {@t{F}{F} = \\exp(-\\tau_{\\rm total})}", "html": "τtotal​(v)=∑i​τi​(v),F=exp(−τtotal​)", "terms": [ "taui", "F" ], "values": [], "box_em": { "w": 21.424, "h": 1.522 } }, "short": { "tex": "{@t{taup}{\\tau_{\\rm peak}} = @v{taup}}", "html": "τpeak​=0.00×100", "terms": [ "taup" ], "values": [ "taup" ], "box_em": { "w": 9.291, "h": 1.522 } }, "spoken": "Optical depths of all cells add, and the transmitted flux F is e to the minus total optical depth. So far the deepest point has optical depth {{taup}}, so the flux there is {{F}} of the light." } } }, "E6": { "id": "E6", "title": "Instrument blur", "svg": "lsf", "beats": [ "B12a" ], "terms": { "R": { "label": "sharpness, resolving power R", "unit": "", "bind": { "src": "spec", "field": "R" }, "fmt": "grp", "prop": "R", "override": "R", "drag": { "scale": "log", "abs": [ 3000, 150000 ], "step": 0.025, "big": 0.25 } }, "fwhm": { "label": "blur width", "unit": "kilometres per second", "bind": { "src": "derived", "fn": "fwhm" }, "fmt": "sig3", "prop": "R" } }, "variants": { "any": { "val": { "tex": "{@t{R}{R} = @v{R}}\\ \\ \\Rightarrow\\ \\ {@t{fwhm}{\\mathrm{FWHM}} = @v{fwhm}\\ \\mathrm{km\\,s^{-1}}}", "html": "R=0.00×100⇒FWHM=0.00×100kms−1", "terms": [ "R", "fwhm" ], "values": [ "R", "fwhm" ], "box_em": { "w": 25.964, "h": 1.522 } }, "sym": { "tex": "{F_{\\rm obs} = F \\otimes \\mathcal{G}},\\qquad {@t{fwhm}{\\mathrm{FWHM}} = \\dfrac{c}{@t{R}{R}}}", "html": "Fobs​=F⊗G,FWHM=Rc​", "terms": [ "fwhm", "R" ], "values": [], "box_em": { "w": 16.743, "h": 2.17 } }, "short": { "tex": "{@t{R}{R} = @v{R}}", "html": "R=0.00×100", "terms": [ "R" ], "values": [ "R" ], "box_em": { "w": 7.899, "h": 1.522 } }, "spoken": "The observed spectrum is the true spectrum blurred by a Gaussian whose full width at half maximum is the speed of light over the resolving power R. At sharpness R of {{R}} the blur is {{fwhm}} kilometres per second wide." } } }, "E7": { "id": "E7", "title": "Noise per pixel", "svg": "noise", "beats": [ "B12b" ], "terms": { "sn": { "label": "exposure, signal-to-noise ratio", "unit": "", "bind": { "src": "spec", "field": "sn" }, "fmt": "sig3", "prop": "sn", "override": "sn", "drag": { "scale": "log", "abs": [ 2, 100 ], "step": 0.025, "big": 0.25 } }, "sig": { "label": "noise per pixel", "unit": "of the continuum", "bind": { "src": "derived", "fn": "sigma_pix" }, "fmt": "fix3", "prop": "sn" } }, "variants": { "any": { "val": { "tex": "{@t{sn}{S/N} = @v{sn}}\\ \\ \\Rightarrow\\ \\ {@t{sig}{\\sigma_{\\rm pix}} = @v{sig}}", "html": "S/N=0.00×100⇒σpix​=0.00×100", "terms": [ "sn", "sig" ], "values": [ "sn", "sig" ], "box_em": { "w": 21.599, "h": 1.522 } }, "sym": { "tex": "{@t{sig}{\\sigma_{\\rm pix}} = \\dfrac{\\sqrt{5}}{@t{sn}{S/N}}}\\qquad {(5\\ \\text{pixels per resolution element})}", "html": "σpix​=S/N5​​(5pixels per resolution element)", "terms": [ "sig", "sn" ], "values": [], "box_em": { "w": 25.541, "h": 3.049 } }, "short": { "tex": "{@t{sn}{S/N} = @v{sn}}", "html": "S/N=0.00×100", "terms": [ "sn" ], "values": [ "sn" ], "box_em": { "w": 9.492, "h": 1.522 } }, "spoken": "The noise per pixel is the square root of 5 over the signal-to-noise ratio per resolution element, with 5 pixels per resolution element. At exposure S over N of {{sn}} each pixel scatters by {{sig}} of the continuum." } } }, "E8": { "id": "E8", "title": "Reading it back", "svg": "aod", "beats": [ "B13" ], "terms": { "chi": { "label": "reduced chi squared of the fit", "unit": "", "bind": { "src": "live", "field": "chi2nu" }, "fmt": "fix2", "prop": "fit" }, "Na": { "label": "apparent column density", "unit": "per square centimetre, log", "bind": { "src": "live", "field": "logNa", "fallback": { "src": "aod", "field": "logN" } }, "fmt": "pow2", "prop": "N" } }, "variants": { "any": { "val": { "tex": "{@t{Na}{N_a} = @v{Na}\\ \\mathrm{cm^{-2}}}\\qquad\\allowbreak {@t{chi}{\\chi^2_\\nu} = @v{chi}}", "html": "Na​=0.00×100cm−2χν2​=0.00×100", "terms": [ "Na", "chi" ], "values": [ "Na", "chi" ], "box_em": { "w": 22.172, "h": 1.522 } }, "sym": { "tex": "{@t{chi}{\\chi^2} = \\sum_j \\dfrac{(F_{{\\rm obs},j} - M_j)^2}{\\sigma_j^2}},\\qquad {@t{Na}{N_a} = \\dfrac{m_e c}{\\pi e^2}\\,\\dfrac{\\int \\tau_a(v)\\,dv}{f\\lambda}}", "html": "χ2=∑j​σj2​(Fobs,j​−Mj​)2​,Na​=πe2me​c​fλ∫τa​(v)dv​", "terms": [ "chi", "Na" ], "values": [], "box_em": { "w": 26.302, "h": 3.146 } }, "short": { "tex": "{@t{Na}{N_a} = @v{Na}\\ \\mathrm{cm^{-2}}}", "html": "Na​=0.00×100cm−2", "terms": [ "Na" ], "values": [ "Na" ], "box_em": { "w": 11.487, "h": 1.522 } }, "spoken": "The fit minimises chi squared, the summed squared difference between the observed flux and the model in units of the noise; the apparent optical depth method turns the area of the dip into a column density. Here N a is ten to the {{Na}} per square centimetre and the reduced chi squared is {{chi}}." } } } } }