Blackbody Radiation Simulator

What Is the Blackbody Radiation Simulator?

How to use the blackbody radiation simulator

1. Start from the state it opens in. The lab boots on Surface temperature T 5772 K, Lower band edge lam1 400 nm, Upper band edge lam2 700 nm and Probe wavelength 550 nm, with the four cards reading 36.64 %, 502.0 nm, 62.94 MW/m² and 23.06 MW/m². Reset returns all four sliders to exactly that.
2. Move the temperature first. Surface temperature T runs from 800 to 12000 K in steps of 1, and it is the only slider that changes the curve rather than the way it is read. Take it up slowly and watch Share of the power inside the band climb, stall and then fall back; the turning point is at 7042 K.
3. Move the two band edges. Lower band edge lam1 runs from 100 to 1400 nm and Upper band edge lam2 from 150 to 3000 nm, both in steps of 1, and together they decide how wide the gold region is. They cannot cross: push one against the other and the lab moves the edge you are not dragging 1 nm clear and rewrites both its slider and its label.
4. Park the probe where you want a reading. Probe wavelength runs from 100 to 3000 nm and drops a thin cream line with a dot where it meets the curve. It feeds M_lam at the probe and Photon energy at the probe and nothing else, so you can move it freely without disturbing a single card.
5. Read the four cards down the right. Share of the power inside the band carries its symbol F; Peak wavelength carries lam_peak = b/T; Total exitance carries sigma T^4, the quantity the Stefan-Boltzmann calculator returns on its own; and Power inside the band is the product of the first and the third.
6. Read the eight cells under the canvas. They give the curve height and the photon energy at the probe, the photon energy at each band edge, the Share below lam1 and Share above lam2, the curve height at the peak, and the Top of the vertical axis.
7. Read the strip headed The frame. It names both numbers the drawing depends on — the top of the vertical axis and how far right the horizontal one runs — and it is rewritten on every update, so it can never describe a frame that is no longer on screen.
8. Watch the frame move, not just the curve. Both axes are rebuilt from the temperature. At 2700 K the horizontal axis runs to 4500 nm and the top of the frame is 1.85e+3 W/m²/nm; at 12,000 K they are 1200 nm and 3.20e+6 W/m²/nm.

The share card and the total exitance card are the pair to watch together. One says where the power goes and the other says how much there is, and between 2700 K and 12,000 K they move in different directions: the share ends lower than it started while the total has climbed from 3.01 MW/m² to 1.18 GW/m². Neither of them is an efficiency, and multiplying one comparison by the other answers no question anybody asked.

Where the 400–700 nm figure comes from in the first place is a different question, and the guide to the electromagnetic spectrum sets out the bands and where the visible one sits among them. This lab starts one step later, from a surface at a temperature, and never asks what colour a wavelength looks.

Worked example: change one thing at a time

Every row below is one of the seven preset buttons, and every cell is a string the running lab printed there. Rows 1 to 6 hold the band at 400–700 nm and move only the temperature slider; row 7 puts the temperature back to 5772 K and moves the two edges instead. Where a cell and the lab ever part company, believe the lab.

Read the share column down and then back up. It rises 5.00, 8.09, 36.64, 38.87, 39.30 and then drops to 26.36 %, which is the one result this lab exists to make visible. Nothing has gone wrong at the bottom: the total exitance column has climbed the whole way, from 3.01 MW/m² to 1.18 GW/m², while the share turned over near the top.

The peak wavelength column explains the turn. At 2700 K the cream tick stands at 1073.2 nm, far to the right of the gold region, and only the leading edge of the curve reaches it. By 12,000 K the tick has swept through the band and out the other side to 241.5 nm, and the region is now catching the curve’s steep left flank on the way down. Between those two the tick crosses the band, and that crossing is where the share peaks.

Row 7 is the one nobody expects. Neither the surface nor its temperature has changed — the peak, the total exitance and the axis maximum are identical to row 1 — and yet the headline moves from 36.64 % to 43.77 % because the gold region has been widened by 20 nm at one end and 50 nm at the other. A percentage quoted without its two edges is not a fact about anything.

The first two columns say where the missing power went. On the filament case almost all of it, 94.93 %, is above 700 nm; on the hottest case 60.75 % has crossed to the other side and sits below 400 nm. Add either row to its headline share and you get 100 %, but that is a definition restated rather than a check on anything: the three cells are a partition of one spectrum by construction.

The two photon-energy columns move only on the last row, because they belong to the edges rather than to the surface. Widening the band to 380–750 nm takes them from 3.0996 and 1.7712 eV to 3.2627 and 1.6531 eV. If you would rather convert a wavelength on its own, the photon energy calculator does it in either direction, and the photon energy guide works through the relation.

The last column is the one this page keeps coming back to. Across those seven rows the top of the frame runs from 1.85e+3 to 3.20e+6 W/m²/nm, a spread of more than three decades, while every one of those curves is drawn to the same height on the canvas. That column, and not the picture, is where the magnitude lives.

Formula and symbol reference

The lab works from three relations and nothing else. The curve is M_lam(lam,T) = 2 pi h c^2 / lam^5 / (exp(hc/(lam k T)) - 1); its peak is at lam_peak = b/T; and its area over all wavelengths is M_total = sigma T^4. The headline share is the area between the two edges divided by that total, and the two share cells are the same quantity taken outside the edges.

One thing in that list is easy to misread. M_lam here is the hemispherical spectral exitance, the power leaving one square metre per nanometre of wavelength, and not the radiance per unit solid angle; the two differ by a factor of pi. The form used is the one whose integral over all wavelengths is exactly the total exitance card, which is what lets the share be a share.

Two rows there carry a formatting rule rather than a physical one. The two power cards switch prefix at a rounded thousand, so 999.996 W/m² prints as 1.00 kW/m² and never as 1000.00 W/m²; below one watt the ladder simply stops, which is why 800 K gives 4.75e-4 W/m² rather than a milliwatt prefix. Every other quantity has one fixed decimal count and no branch at all, so a screenshot taken today reproduces exactly.

The physics: a lopsided curve inside a frame that keeps moving

Two things on this canvas are doing the work, and only one of them is the curve. The curve is lopsided: it climbs almost vertically on the short-wavelength side and falls away over thousands of nanometres on the long side, which you can see at a glance on the filament case where the gold region is a sliver on the steep flank and the whole tail lies to the right of it. The frame around it is rebuilt from the temperature on every update, which is the part that can mislead.

The asymmetry is why the obvious guess about the best temperature is wrong. Set the temperature to 5269 K and the Peak wavelength card reads 550.0 nm, exactly halfway between the two edges, with the cream tick standing in the middle of the gold region — and the share reads only 33.71 %. Load The best temperature there is instead and the tick jumps to 411.5 nm, hard against the left-hand edge, while the share climbs to 39.30 %.

Centring the peak looks fair and is not, because the two sides of the curve are not the same shape. A peak in the middle throws the long slow tail out past the upper edge where it is not counted, while pushing the peak towards the lower edge loses only a little off the steep side and drags that tail inside the band. the full guide to blackbody radiation works that trade through with the arithmetic beside it.

Now the frame. The vertical axis is set to the peak curve height at whatever temperature is loaded, so the apex always touches the top and the number printed there is a value the drawing genuinely reaches. That is honest about the shape and silent about the size: at 2700 K the top of the frame is 1.85e+3 W/m²/nm and at 12,000 K it is 3.20e+6, yet both curves are drawn the same 403.0 px tall. That peak height goes as the fifth power of the temperature, so the ratio between those two frames is (12000/2700)^5 = 1,734, which is not what dividing the two rounded figures above would give you.

The horizontal axis moves too, and rather more visibly. It runs to four times the peak wavelength, clamped between 1200 and 6000 nm and rounded to a tidy tick, and it is raised further if it would otherwise clip the gold region — so the shaded band is never cut off at the right-hand edge. Across the seven presets it takes six different values, from 1200 nm on the hottest to 4500 nm on the filament.

That is why the caption along the bottom of the canvas says what it says. It reads Vertical scale is set by this temperature — compare shapes, not heights, and it is assembled from the lines actually painted rather than stored as a constant, so a caption that failed to fit cannot be reported as though it were on the screen. The strip headed The frame under the canvas repeats both live numbers in prose.

Where the blackbody radiation simulator breaks down

The lab solves its own model exactly, so nothing on the screen ever fails. Everything below is a limit of that model, of the figures you feed it, or of what the drawing can carry, and each item says what this lab does about it.

Curve heights cannot be compared between two temperatures

The vertical axis is rebuilt at every update, so the apex always touches the top of the frame whatever the temperature. That keeps a 2700 K curve visible beside a 12,000 K one and makes their drawn heights meaningless against each other. Take the magnitude from the Top of the vertical axis cell or from the two power cards; take the shape from the canvas.

The horizontal axis moves as well, and by more than it looks

The frame runs to four times the peak wavelength rather than to a fixed wavelength, so the gold region occupies a different fraction of the picture at every temperature even when its two edges have not moved. On the filament case the band covers a narrow strip of a 4500 nm axis; on the hottest case the same band covers a quarter of a 1200 nm one. Read the tick labels before reading the width.

A card reading 0.00 % is not a card reading zero

The three share readouts carry two decimals, so anything under half a hundredth prints as 0.00 %. At 1000 K the share between 400 and 700 nm does exactly that, while the Power inside the band card beside it reads 1.04e-1 W/m² — a small number with real power behind it. The right sentence is that the share is under a thousandth of one per cent, never that it is nothing.

Below one watt the prefix ladder stops rather than inventing a smaller one

The two power cards step through kilowatts, megawatts and gigawatts, and the prefix is chosen from the figure after it has been rounded, so 999.996 W/m² prints as 1.00 kW/m². Going the other way there is no milli or micro rung: at 800 K the band card prints 4.75e-4 W/m² in exponential form. That is a display decision, not a limit on the model.

The peak marker and the probe line are dropped by a pixel test

The peak tick and its label are skipped whenever the canvas is narrower than 380 CSS pixels, and whenever the label would come within 14 pixels of a band edge label; the probe goes when its own label cannot be placed or when it falls outside the frame. Both cards go on printing their figures regardless. A dropped label always takes its line with it, so the canvas never carries a guide nobody can name.

Both edges are sliders because neither of them is fixed

Where the visible slice starts and stops is a stated convention, and 400–700 nm is what this site states. Nothing measures it, which is why the lab hands you both edges instead of printing them as furniture: drag them out to 380 and 750 nm and the 5772 K surface goes from 36.64 % to 43.77 % without the surface changing at all. No percentage read off this page means anything without the pair of edges beside it.

The cream tick belongs to the curve on screen and to no other

What it marks is the highest point of power per unit wavelength, which is the only quantity this canvas plots. Redraw the identical physics against frequency and the highest point lands somewhere else on the spectrum, and no conversion between the two variables will carry one onto the other, because the change of variable does not stretch the axis evenly. Read the tick as the peak of the per-wavelength curve, then, and never as the place where a 5772 K surface simply peaks.

There is no emissivity slider, and that is deliberate

Every curve on this canvas is an ideal radiator’s. Put a real material in its place and each wavelength comes out weakened by a factor of its own, so the measured curve is a different shape rather than a shorter copy of this one. A single flat factor across the whole spectrum would divide straight out of the share and leave the drawing untouched, which would hide the one thing worth knowing about a real emitter. Where the total is the question, the Stefan-Boltzmann calculator carries that term and the Stefan-Boltzmann lab draws the fourth-power law.

The three shares add to 100 % because they are defined that way

The headline share and the two cells either side of it are a partition of one spectrum, so watching them total 100.00 % confirms nothing about the lab. The same goes for the power inside the band being the share times the total, which is the definition of the share rearranged. Nothing on this page is checked by any other number on this page.

The two edges cannot cross, and the lab moves one rather than refusing

If a drag would leave the two edges level or the wrong way round, the edge you are not dragging is pushed 1 nm clear and written back into both its slider and its label, so no impossible band ever reaches a readout. Drag the upper edge down to its 150 nm minimum and the lower edge is carried down to 149 nm with it, where the share is 0.00 % and the band card reads 287.58 W/m². That is a real 1 nm band, not an error state.

Nothing under the gold region is coloured

Mapping a wavelength onto a screen colour needs conventions this lab has not adopted, and a rainbow there would say the page is about colour when it is about power. The shading marks which slice is being counted and claims nothing else. What a prism does with that slice is the dispersion of light lab, and the bands either side of it are the electromagnetic spectrum lab.

Nothing here has been measured

The temperature and both band edges are figures you choose, and no reading describes a particular lamp, star, filament or instrument; the preset names are shorthand for the temperatures beside them. 5772 K is the IAU nominal solar effective temperature rather than a reading taken from the Sun, and every solar figure here is top-of-atmosphere — sunlight at the ground has already lost most of its ultraviolet. Verify anything you intend to rely on against your own data first.

Where the blackbody spectrum is actually used

Separating a share from a quantity, on one screen

The commonest error with thermal radiation is treating “more of it is visible” and “more of it comes out” as one statement. Here they are two cards that plainly disagree: between the filament case and the hottest case the share ends lower than it started while the total exitance has gone from 3.01 MW/m² to 1.18 GW/m². Neither figure is lumens per watt and neither is an efficiency.

Choosing the window a non-contact thermometer looks through

An instrument that infers temperature from brightness sees only a slice of the spectrum, and how steeply that slice brightens with temperature is what sets its sensitivity. Set the two edges to the instrument’s window, step the temperature through the range you care about and watch the Power inside the band card rather than the share. The ratio of two of those readings is the signal you would actually get.

Finding where a threshold device stops responding

Anything with an energy threshold responds only to wavelengths below some edge. Park the probe on that wavelength and the Photon energy at the probe cell gives it in electronvolts on the spot; set the upper band edge to the same place and the share card gives the fraction of the incoming power that is usable at all. Both readings come from one setting of the sliders.

Seeing why two surfaces at different temperatures barely overlap

Load a surface near 5772 K, note where the frame puts the curve, then drop the temperature to a few hundred kelvin above room temperature and watch the whole picture slide right while the axis maximum collapses. Absorbing across one of those bands and re-radiating across the other is the whole idea behind a wavelength-selective coating, and the separation is visible before any arithmetic is done.

Reading somebody else’s spectrum plot without being fooled

Published Planck curves are very often normalised, and two of them side by side can be drawn to the same height on purpose. This lab does the same thing and says so in three places, which makes it a fair demonstration of the habit. Once you have watched the frame move here, the first question to ask of any such plot is what its vertical axis was set from.

Setting a question that cannot be answered by guessing

“Which temperature puts the most visible light into 400 to 700 nm?” has a wrong answer that almost everyone reaches first, and the sliders settle it in about ten seconds. The follow-up is better still: ask for two different temperatures that give the same share, and the shape of the temperature sweep does the explaining. The blackbody radiation calculator will not solve backwards for temperature for exactly that reason.

Where to go next

For the argument in full, with the worked problems, the three-way split table and the diagrams that go with them, read Blackbody Radiation: Why 39% Is the Best Light You Get. If you would rather type figures than drag them, the blackbody radiation calculator takes the same three inputs and also runs the question backwards, returning the band edge that captures a share you name; it prints six significant figures where this lab prints two decimals, so its 36.6383 % and this lab’s 36.64 % are the same number.

The two ends of this calculation have tools of their own. The Stefan-Boltzmann law owns the total exitance card, with its calculator and its simulator beside it, and the Wien law calculator returns the peak wavelength card on its own for any temperature.

Nearby, the electromagnetic spectrum guide places the visible slice among the other bands with its lab, the guide to dispersion of light and the dispersion lab take that slice apart, and the wavelength formula and units guide covers nanometres and the rest of the scale. The photon energy guide and its calculator handle the edge cells, and the guide to absolute zero with its lab and its calculator covers the other end of the temperature slider. The rest is in the library of physics simulations and on the blog, and the site search will find a topic by name.