# At what point does a line spectrum become a continuous one?

I have 1 main and 1 sub question, please dont remove my post for this i just want to clear this up its been bugging me, thanks.

Fundamentally, I don't understand 2 things, and if anyone could clear this up I would really appreciate it; at what point does an emission line spectrum become continuous? Does a human for example have a continuous or line spectrum? why/why not? I know hydrogen has a line but blackbodies have continuous.

Also, what determines at what intensity (flux) I will emit at each wavelength? Does the flux for continuous spectra always look similar to the blackbody one:

Like if I heated an iron rod to 500 $$^\circ$$C would it look like above? Perhaps, and I find this most convincing, this is a statistical thing. Just as the normal distribution pops up for rolling a die hundreds of times, the same is true here, just instead of dice there are atoms releasing and absorbing energy and instead of a normal distribution it looks like above.

• heat radiation (continuous) $\neq$ stimulated radiation (discrete for atoms and molecules) Apr 16 at 15:30
• @Mauricio I dont know what those are and why Apr 16 at 15:48
• youtube.com/watch?v=-lHXZk5M6cI gives a nice "quantum physics for dummies" explanation of energy levels and how discrete lines multiply and merge when particles interact. It's about solid-state semiconductors rather than stars, and focuses on electrical conductivity with photons just being a necessary step along the way, but still, it covers some of the material you're missing in an accessible way. Apr 17 at 0:00

The easy (and somewhat oversimplified) answer is: you get a continuous spectrum once you have so many lines that you can no longer tell them apart.

The more precise answer is that the electrons in a single atom exist in bound states, which means their energy levels are quantized, meaning discrete. To transition between two energy levels, they need to either absorb a photon with a wavelength $$\lambda$$ corresponding to the energy difference $$\Delta E$$ between the levels, or emit a photon which again has a wavelength corresponding to the energy difference. The relation between $$\Delta E$$ and $$\lambda$$ is: $$\Delta E = \frac{hc}{\lambda}$$ where $$h$$ is Planck's constant and $$c$$ is the speed of light. Now since only a discrete set of energy levels is allowed, the atom can only emit a discrete spectrum of light. The more atoms you add however, the more complicated the system becomes, and more energy levels become accessible due to interactions between the atoms. In thermodynamics and statistical mechanics, one does not consider single atoms, but rather large ensembles of particles. In this case, it no longer makes sense to consider the emission lines of single atoms, and we model the the spectrum as continous.

In that sense, your intuition about this being a statistical effect is correct: The spectrum now essentially becomes a probability density for the energy of a single particle in your ensemble. I assume that the spectrum of a heated iron rod would closely resemble a blackbody spectrum. If you had a spectrometer with infinite resolution, you would see lines in every blackbody spectrum, but this is of course unphysical.

Generally, the spectrum of a radiating body is determined by its energy and chemical structure - atoms which are tightly bound in a crystal lattice or molecule will have different probability densities for their energies than those in a gas, for example.

• Thank you, this is a good answer. Just some follow up questions: If particles in bulk produce a continous spectrum of emission (that means they absorb and emit all wavelengths), doesn't that mean that all objects are black bodies in some capacity? Also, why does the flux/wavelength graph for black bodies look the way it does? And if all objects are blackbodies then wouldn't all objects exhibit a similar graph? Apr 16 at 20:24
• The plots in the question refer to stars. The explanation presented here does not apply to stars. Apr 16 at 20:30
• @ProfRob yes but that shape applies to black bodies in general as well, loosly speaking. The lower the temperature the lower the peak and slightly flater curve etc. Apr 16 at 20:42

The spectrum plots you show in your question are claimed to represent stars of different temperatures. Stars do indeed show a continuous spectrum and the reasons for this depend on the effective temperature of the star's photosphere - the layer from which radiation is able to escape.

In sun-like stars, the continuum radiation is caused by a radiative (re)combination between free electrons and hydrogen atoms, forming a hydride ion (H$$^{-}$$). Since the free electrons occupy a continuum of energy states, the photons emitted when they combine with hydrogen have a continuous spectrum.

In hotter stars, the continuous radiation is also formed by recombination continua as free electrons combine with protons (ionised hydrogen).

In cooler stars, there is very little true continuum. What appears to be a continuum is actually a densely packed array of molecular ro-vibrational transitions.

In heated metals and other solids, something similar occurs where atoms and molecules behave collectively to form bands of energy states and transitions between these are essentially a continuum.

A more general answer to your question is that for bodies that emit radiation that looks a bit like an idealised blackbody continuum, the emission processes are just the opposite of the processes that would absorb photons over a continuum of frequencies. If such absorbing processes don't exist, then the body would essentially be transparent at those wavelengths and neither would it emit at those wavelengths and could not be a blackbody.

As for what determines the intensity of the received spectrum - in the case of something approximating a blackbody, it is simply the temperature and the solid angle (area divided by the square of distance) of the emitting object. The shape of the spectrum is the Planck distribution.

Line spectrum comes from single atoms in gases , band spectrum from molecules and continuous spectrum from heated solid bodies.

• So do humans emit a continous spectrum similar to that of the black body one in the post? Apr 16 at 15:49
• @user15755358 yes humans emit a continuous spectrum Apr 16 at 15:58
• Theoretical yes, but consider that the temperature is about 310°K so far in infrared. Apr 16 at 15:59
• this is starting to become more clear for me, but why do atoms have line spectrums but groups of atoms have continuous spectrum, do the vibrations or something cotribute? Apr 16 at 16:04
• @trula so could a human, and everything else be considered a black body albiet bad ones? Since they have a similar emission spectrum/intensity graph as a black body? Apr 16 at 16:06