# Is there a fundamental limit to the temporal resolution of signals from space?

In Earth-based experiments, we can measure phenomenon very rapidly in an experiment given appropriate equipment. Clearly if something takes a long exposure to see (due to a weak signal), then the temporal resolution will be filtered by the exposure time. But let's assume a signal was strong enough that exposure time wasn't a concern.

When conducting observations of things like transits around stars, tracking orbits of moons in our own solar system, or watching the evolution of supernovae, is there a fundamental limit to the temporal resolution? Does the distance introduce aliasing in the signal, perhaps due to differences in wave speed and packet speeds that only become evident over large distances?

• Have you read up on millisecond pulsars? These are faint and fast. – DarenW Aug 29 '13 at 6:49

## 2 Answers

There are three ways I can imagine time resolution being limited: integration time, dispersion, and intrinsic width.

# Integration Time

One of these ways you already eliminated: Dim sources will need longer integration times to overcome Poisson statistics/read noise/etc. in order to trigger any detection at all.

# Dispersion

## Theory

The second way is the most direct. As discussed in another answer, a pulse will broaden in time in a dispersive medium. While there will be no dispersion in perfect vacuum,1 the interstellar medium (ISM) does have free charges, and so is dispersive to some extent.

A good discussion of this can be found in any text on the ISM, such as [1], whose salient points I will expand upon here. The dispersion relation between wavenumber $k$ and frequency $\omega$ (in radians per second) for a plasma is given by $$c^2 k^2 = \omega^2 - \omega_\mathrm{p}^2,$$ where $\omega_\mathrm{p} = \sqrt{4\pi n_\mathrm{e} e^2/m_\mathrm{e}}$ is the plasma frequency (in CGS!) in terms of electron number density $n_\mathrm{e}$, electron charge $e$, and electron mass $m_\mathrm{e}$. The group velocity is $$v_\mathrm{g} = \left\lvert \frac{\mathrm{d}\omega}{\mathrm{d}k} \right\rvert = c \sqrt{1 - \left(\frac{\omega_\mathrm{p}}{\omega}\right)^2}.$$

The travel time across a distance $D$ is simply $$t = \int_0^D \frac{1}{v_\mathrm{g}} \, \mathrm{d}x.$$ Now since $\omega_\mathrm{p}$ corresponds to (circular) frequencies $\nu$ of less than $10\ \mathrm{kHz}$ at electron densities of $1\ \mathrm{cm}^{-3}$, we often have $\omega \gg \omega_\mathrm{p}$.2 With this approximation, we can write $$t \approx \frac{1}{c} \int_0^D \left(1 + \frac{1}{2} \left(\frac{\omega_\mathrm{p}}{\omega}\right)^2\right) \, \mathrm{d}x = \frac{D}{c} + \frac{2\pi e^2}{cm_\mathrm{e}\omega^2} \mathrm{DM},$$ where $$\mathrm{DM} = \int_0^D n_\mathrm{e} \, \mathrm{d}x$$ is known as the dispersion measure along the line of sight. The spread in arrival times is then seen to obey $$\Delta t \approx -\frac{4\pi e^2\mathrm{DM}}{cm_\mathrm{e}} \omega^{-3} \Delta\omega = -\frac{e^2\mathrm{DM}}{\pi cm_\mathrm{e}} \nu^{-3} \Delta\nu. \tag{1}$$ Here the negative sign indicates higher frequencies will arrive earlier.

## Plugging in Numbers

A typical line of sight terminating in our own galaxy will have a dispersion measure of something like $100\ \mathrm{pc}/\mathrm{cm}^3$. If you are dealing with sources in other galaxies, there will be a similar contribution from the other galaxies' ISM, as well as a contribution from the intergalactic medium. This latter value can exceed $1000\ \mathrm{pc}/\mathrm{cm}^3$ for very distant galaxies.3

With this in mind, we can rewrite (1) to be $$\Delta t \approx (0.8\ \mathrm{s}) \left(\frac{\mathrm{DM}}{100\ \mathrm{pc}/\mathrm{cm}^3}\right) \left(\frac{\nu}{1\ \mathrm{GHz}}\right)^{-2} \frac{\Delta\nu}{\nu}. \tag{2}$$ Clearly lower frequencies will suffer more dispersion.

In the millimeter/submillimeter regime, an array like ALMA might record data in the $\nu = 110\ \mathrm{GHz}$ band with a resolution of $\nu/\Delta\nu = 3\times10^{7}$. A single channel will therefore be broadened in time by $\Delta t \approx 2\times10^{-12}\ \mathrm{s}$ in this case, small enough that integration time and other effects will overwhelmingly dominate.

It is worth mentioning that even millisecond pulsars' pulses are much more spread out in frequency and time than the above numbers. For a thorough review of millisecond pulsars, see [2]. In particular, Figure 6 from that article shows a pulse from pulsar B1356-60 (period $128\ \mathrm{ms}$, $DM = 295\ \mathrm{pc}/\mathrm{cm}^3$) as detected in nearly $100$ channels from $1.24\ \mathrm{GHz}$ to $1.52\ \mathrm{GHz}$. Across the whole band, the pulse is dispersed to have a width of nearly half a second. Each channel, however, sees a pulse lasting about $10\ \mathrm{ms}$. Thus we see that higher spectral resolution combats dispersion.

How far have we pushed this technique? In [3] the authors report resolving the pulse structure of the Crab pulsar's emissions down to the nanosecond timescale. (Figure 21 in [2] shows these results.) They mention two other potential sources of pulse broadening - Faraday rotation and path length differences due to scattering - but say that these are not limiting. To overcome the difficulty of building an instrument with very high spectral resolution as (2) would seem to imply, they pass the signal through a digital filter with a transfer function designed to negate dispersion, the so-called "coherent dispersion removal technique."

# Intrinsic Timescales

The third way in which useful data has a time resolution limit revolves around the fact that there simply may be no significant changes on short enough timescales. Part of why [3] was considered interesting is that one has to push theories pretty hard to come up with a way that the bulk of an astrophysical object can undergo coordinated changes so quickly. After all, even light will take several microseconds at least to cross the diameter of a neutron star, and anything energetic and smaller than a neutron star tends to collapse to a black hole.

In fact, it is most often the case in astrophysical systems that some observed event happens as soon as it knows that it can happen. Put another way, only somewhat contrived and unlikely physical systems can distribute information internally and then only later respond in a dramatic fashion to that information all at once.

Putting this principle to use, one can see that the timescales worth probing will tend to get longer as the sizes of the sources get larger. One such example is found in reverberation mapping of active galactic nuclei. In that technique, the clouds orbiting the central supermassive black hole and its accretion disk are lit up by flares of activity in the central engine. These clouds (comprising the broad-line region) orbit at roughly $100\ \mathrm{AU}$, and the geometry of the situation means different clouds are seen by us to light up at different times. There are many fascinating things you can do with this information (see [4] for a review), but the take-away message is that sometimes things simply take a long time to change, especially when they are so far away that we can only see them because they are large.

# Conclusion

In many cases our time resolution is only limited by equipment one way or another. But in certain regimes there is enough dispersion to be noticed, and there is a fast enough source to care, and so we push for higher resolution. As a result, there are systems whose nanosecond timescales have been observed.

Finally, regarding some of the examples given in the question:

• Exoplanet transit timescales used to be set by the duration (orbital speed divided by stellar diameter, or half a day for Earth). Now astronomers regularly measure ingress and egress, which last as long as the planet takes to move its own diameter (several minutes for Earth). Phase folding is the cheat that makes this easy -- if you get to keep re-watching identical replays of the same event, there really isn't any limit to your time resolution.4
• While the final burning, collapse, and/or bounce phases in supernovae may very well happen on sub-second timescales, such things are not directly visible. Instead we only see these explosions long after they have gone off. Typical timescales are days to weeks for watching these events decay. If you happen to catch one very early (before maximum brightness but still after the initial explosion), you might see noticeable variation on the timescale of hours.
• It is interesting that moons were brought up in this discussion of "time-domain astronomy" as it is called. I would argue that the first modern time-domain astronomical observation was Rømer's determination of the speed of light. By watching when Io was eclipsed by Jupiter (actually, when Io entered and left Jupiter's shadow, as this is a very sharp transition) over a long period of time, one can precisely predict when the next such event will occur. However, the varying distance between Jupiter and Earth results in the eclipses appearing up to several minutes early or late.

1 Certain attempts at quantum gravity predict vacuum propagation speeds of photons that depend on energy. Very tight constraints on this violation of Lorentz invariance have been obtained with astronomical observations. For instance, in [5] it is reported that the detections of photons over a very broad range of energies from a gamma ray burst were coincident within a second. Given that those photons were traveling for well over $7$ billion years, this shows observationally that such effects can be at most tiny.

2 In fact, frequencies below the plasma frequency will exponentially attenuate rather than propagate through the ISM, so one cannot even observe them.

3 I discussed the reverse problem, using pulse widths to find dispersion measures and hence distances, in an answer to another question.

4 This is the exact same method for achieving incredibly high time resolution in "femto-photography."

[1] Bruce T. Draine. Physics of the Interstellar and Intergalactic Medium. Princeton, 2011.

[2] Duncan R. Lorimer. "Binary and Millisecond Pulsars." Living Reviews in Relativity, 11, 2008.

[3] T. H. Hankins et al. "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar." Nature, 422, 2003.

[4] B. M. Peterson and K. Horne. "Echo mapping of active galactic nuclei." Astronomische Nachrichten, 325, 2004. (preprint here)

[5] A. A. Abdo et al. "A limit on the variation of the speed of light arising from quantum gravity effects." Nature, 462, 2009. (preprint here)

I can comment only on the last part of your question. From classical electromagnetic wave perspective, the question whether aliasing is introduced to the signal or not depends on the medium through which the wave is travelling. There are two categories of media according to this:

1.Non-dispersive media: in this type of media, there is no difference between wave speed and packet speed. They are both equal to the speed of light. This type of materials can be described mathematically by permittivity and permeability that both do not depend on frequency.

2.Dispersive media: in this type of media, there is a difference between wave speed and packet speed. The different frequency components in the packet travel at different speeds causing the packet signal to become distorted and spread in time. This type of materials can be described mathematically by frequency dependent permittivity and permeability.

As an example, say you have a pulsed signal consisting of three pulses separated by off times. If that signal travelled through a non-dispersive media. The signal theoretically will maintain its temporal characteristics without any change, regardless of distance it travels.

However, if the same pulse travelled through a dispersive media, the signal will gradually transform from being well defined on and off periods to a continuous signal. In such a case, the distance a wave travels matters, the larger the distance is, the more dispersion is experienced and the more the signal becomes distorted.

In space, vacuum is non-dispersive, while plasma is dispersive. So if a signal is coming from a distant place in the universe, the path it travels determines whether it will be distorted or not. And if it was distorted, how significant the distortion is.

As I mentioned at the beginning, this explanation is based on classical electromagnetic wave perspective, it doesn’t account for quantum or gravitational effects.

Hopefully that was useful