**2.1. Calculation of the minimum-energy field**

The magnetic field strength and particle spectrum are important for jet physics as they define the internal pressure. The level of synchrotron radiation depends on the magnetic-field strength and the number of relativistic electrons and positrons, but these quantities are inseparable based on the observed synchrotron radiation alone. To progress further it is usual to assume that the source is radiating such that its combined energy in relativistic particles and magnetic field is a minimum [35]. In this situation the energy in the magnetic field is ~ 3/4 of the energy in the relativistic particles, and so this is similar to the condition in which the two are equal and the source is in `equipartition'. A change in any direction of the ratio of energy density in particles to magnetic field increases the total energy and pressure in the emitting plasma.

The minimum-energy magnetic field for a power-law spectrum of
electrons producing radiation of a measured flux density at a
particular frequency can be calculated analytically (e.g.
220),
and for more complicated spectra the
results can be obtained via numerical integration.
Physical insight can be gained by considering a
power-law spectrum where electrons give rise to a synchrotron luminosity,
*L*_{},
at a given frequency of the form

(1) |

It is now normally
thought preferable to define the spectral limits via a minimum and
maximum Lorentz factor for the electrons in the source frame,
_{min}
and _{max}, (e.g.
220),
rather than as synchrotron frequencies in the observer's frame (e.g.
147),
since the former is related to acceleration processes and has the
potential for being chosen on a physical basis. Except in the special
case of = 0.5,
the minimum-energy magnetic field strength, *B*_{me}, is
given by

(2) |

where *V* is the source volume, and
*C*_{1} and *C*_{2} are combinations of
fundamental physical constants and
functions of given by
synchrotron theory (for details see
[220]).
Following the notation of
[147],
*K* is the ratio of energy in other relativistic
particles to that in the electron and positron component,
and is the
fraction of the
volume filled by particles and fields (the so-called filling factor).
The true minimum energy is when the only relativistic particles are
radiating leptons, and the volume is completely and uniformly
filled with radiating particles and fields. Some authors
consistently use these assumptions when calculating *B*_{me}.
If *K* > 0 or
< 1 then
*B*_{me} is increased.
Results for *B*_{me} are more strongly dependent on
_{min}
than _{max}, since
> 0.5
for most observed radio spectra.

Relativistic beaming of a source affects *B*_{me} (as
considered later in Section 3.2). Since
there is inevitably uncertainty in the value of beaming parameters,
*B*_{me} is best measured in components for which
bulk relativistic motion is believed to be small or negligible.
Of course, even in the absence of relativistic beaming, the angle to the
line of sight, , enters
into the calculation via a correction from
projected linear size into true source volume, *V*.
Typical values found for *B*_{me} in radio lobes and
hotspots are 2-200 *µ*Gauss (0.2-20 nT) (e.g.
119),
although a hotspot field as large as 3000 *µ*Gauss has been
measured
[91].

Figure 3 shows the dependence of
*B*_{me} on
_{min},
*K*, , and
,
separately for electrons giving rise to synchrotron spectra
with = 0.6 and
= 1.1. The former slope is as expected from electrons
undergoing highly relativistic shock acceleration
[1],
and the latter where energy losses have steepened the spectrum.
The curves show that *B*_{me} changes rather little (within
factors of at most a few) for rather large changes in the input
assumptions.

**2.2. Using X-rays to test minimum energy**

The minimum-energy assumption can be tested by combining measurements of synchrotron and inverse-Compton emission from the same electron population. If the inverse Compton process is responsible for most of the X-ray radiation that is measured, and the properties of the photon field are known, the X-ray flux density is proportional merely to the normalization of the electron spectrum, , if the usual power-law form

(3) |

is assumed, where *N*_{e}^{(rel)} is the number of
relativistic electrons per unit
. The
upscattered photons
might be the CMB, whose properties are well known. Alternatively they
could be the radio synchrotron radiation itself, in the process known
as synchrotron self-Compton (SSC), or photons from the active nucleus,
particularly at infrared through ultraviolet frequencies. Since the
available photons range in frequency, so too do the energies of
electrons responsible for scattering them into the X-ray, and these
are rarely the same electrons for which the magnetic field is probed
through synchrotron radiation. Nevertheless, it is usual to assume
that the magnetic field, photons, and relativistic electrons are
co-located, with the synchrotron photon density proportional to
*B*^{1+}.
Here is defined as in
Equation 1, and theory gives
= (*p* - 1) / 2.
The combination of synchrotron
(radio) flux density and inverse Compton (X-ray) flux density then
allows a value for the magnetic field strength, *B*_{SiC},
to be inferred and compared with *B*_{me}.

Since the modelling requires that the volume and any bulk motion of the emitting plasma be known, the best locations for testing minimum energy are the radio hotspots, which are relatively bright and compact, and are thought to arise from sub-relativistic flows at jet termination (but see [80]), and old radio lobes where the plasma may be relatively relaxed. There is no reason to expect dynamical structures to be at minimum energy.

It was anticipated that *Chandra* and *XMM-Newton* would make
important advances in tests of minimum energy, since already with
*ROSAT* and *ASCA* there were convincing detections of inverse
Compton X-ray emission from the hotspots and lobes of a handful of
sources (e.g.
101,
76,
195),
and pioneering work on the hotspots of Cygnus A had found good agreement with minimum energy
[101].
*Chandra* and *XMM-Newton* have
allowed such tests to be made on a significant number of lobes and
hotspots, with results generally finding magnetic field strengths
within a factor of a few of their minimum energy (equipartition)
values for *K* = 0 and
= 1 (e.g.
95,
33,
110,
51,
30,
10,
54,
148).
A study of ~ 40 hotspot X-ray
detections concludes that the most luminous hotspots tend to be in
good agreement with minimum-energy magnetic fields, whereas in
less-luminous sources the interpretation is complicated by an
additional synchrotron component of X-ray emission
[98].
Considerable complexity of structure is seen where hotspots are close
enough for X-ray images to have kpc-scale or better resolution (e.g.
126).

For radio lobes,
the largest systematic study where it is assumed
that all the X-ray emission is inverse Compton radiation is of 33 FRII
lobes, and finds 0.3 < *B*_{SiC} / *B*_{me}
< 1.3
[55].
Since the asymmetry is on the side of *B*_{SiC} <
*B*_{me}, it is important to recognize that the analysis
may not have accurately taken into account contributions to the
lobe X-ray emission from cluster gas, now commonly detected away from
the lobe regions in FRII radio galaxies
([12],
and see Fig. 2). However,
as seen in Figure 4, the lobe X-ray emission
from cluster gas would have
to be far brighter than that from inverse Compton scattering to cause
*B*_{SiC} / *B*_{me} to increase significantly
(e.g., from 0.5 to 1.0), and this is incompatible with the observation
that lobes stand out in X-rays as compared with adjacent regions.

Better agreement between *B*_{SiC} and
*B*_{me} would be
achieved if *B*_{me} has been overestimated.
Figure 3
shows that decreasing the filling factor or including relativistic
protons that energetically dominate the electrons
have the opposite effect. A decrease in *B*_{me} is found
if the source has been assumed to be
in the plane of the sky whereas it is really at a small angle, with
the structures having more volume. However, the small angles required
to make an appreciable difference would be inconsistent with random
sampling. More promising would be if
_{min}
were higher than typically assumed, as stressed by
[27]
who claim evidence for a value of
_{min}
as high as ~ 10^{4} in
the hotspot of one FRII radio galaxy, with a lower value of
_{min}
~ 10^{3} in the lobes as a result of adiabatic expansion. This
is in line with earlier measurements of spectral flattening at low radio
frequencies in hotspot spectra, suggestive of values of
_{min} no lower than a few hundred (e.g.
133,
41).
Why there might be such a
_{min} in a hotspot is discussed by
[91].

It is important to stress that finding *B*_{SiC} /
*B*_{me} within a factor of a few of unity does not allow
strong constraints to be placed on physical parameters. As shown in
Figure 3,
large changes in input parameters do not change *B*_{me},
and thus *B*_{SiC} / *B*_{me}, by a large
amount. It is often pointed out that if the magnetic-field strength is a
factor of a few below *B*_{me}, the energy in relativistic
electrons must dominate the
magnetic-field energy by orders of magnitude. While this is relevant
for understanding the state of the plasma, does this really matter
from the point of view of source energetics? The increase in combined
electron and magnetic-field energy over the minimum energy is
relatively modest as long as the electron spectrum is not very steep
and the field strength is no less than about a third of
*B*_{me} (Fig. 5).

In any case, it is clear that application of minimum energy over large regions is an oversimplification. Three-dimensional magneto-hydrodynamical simulations that incorporate particle transport and shock acceleration [201, 202] find much substructure of particle distributions and fields within the volumes typically integrated over observationally. Complexity on a coarser scale is seen in some observations (e.g. 110, 148).