<p>It would mean that the kinetic energy density of the electrons was equal to the energy density of the magnetic field.</p>

CC BY-SA 4.0
2021-12-18T21:59:52.177Z
Oh my goodness I've parsed the sentence differently than you, that's the problem! "X from Y in equipartition with Z"  I read it as "X (from Y) in equipartition with Z" but it's "X, from Y *which is* in equipartition with Z". Yes this solves it. *Thanks!*
uhoh
CC BY-SA 4.0
2021-12-18T17:31:50.230Z
2021-12-18T17:31:50.230Z
ProfRob
<p>I've heard of the <a href="https://en.wikipedia.org/wiki/Equipartition_theorem" rel="nofollow noreferrer">equipartition theorem</a> in the context of thermodynamics.</p>
<p>But I don't understand how either &quot;synchrotron emission&quot; or &quot;relativistic electrons&quot; can be in &quot;equipartition with a ∼1.5 mG magnetic field&quot;.</p>
<p>Does it mean that the energy density stored in the magnetic field is equal to (or is equilibrated with) the energy density of the relativistic electrons and/or the synchrotron radiation that they produce?</p>
<p>The phrase is from a block quote in <a href="https://astronomy.stackexchange.com/a/43198/7982">this extensive &amp; thorough answer to Nature and mechanism of the short-term variability in radio strength of Sgr A*?</a> citing F. Yusef-Zadeh (2017) <a href="https://arxiv.org/abs/1706.02750" rel="nofollow noreferrer">ALMA and VLA observations of emission from the environment of Sgr A*</a></p>
<p>I see &quot;energy density&quot; half-way through the paragraph below, so I think I may be on the right track, but this is over my head:</p>
<blockquote>
<p>4 DISCUSSION</p>
<p>4.1 Low Extinction Millimeter Halo</p>
<p>[...]The millimeter halo is coincident with the X-ray emission around Sgr A*, which is dominated by bremsstrahlung arising from a medium with n<sub>e</sub>∼150 cm−3 and T∼3×10<sup>7</sup> K. The bremsstrahlung contribution at 230 GHz, about 0.2 µJy, is negligible. Thermal continuum from dust can also be ruled out because of the halo’s extinction deficit of 0.5 magnitudes at H-band relative to its surroundings. The millimeter emission could, however, be produced by synchrotron emission from relativistic electrons in equipartition with a ∼ 1.5 mG magnetic field. The energy density of each of these components would then be ∼ 10% of the thermal energy density of the hot gas, so this is plausible. The luminosity in the mm is <span class="math-container">$4 \pi d^2 \nu S_{\nu}$</span> ∼1.4×10<sup>33</sup> erg s<sup>−1</sup>, comparable to the X-ray luminosity, LX ∼1×10<sup>33</sup> erg s<sup>−1</sup>, implying that synchrotron cooling is marginally the dominant cooling mechanism for the gas. The synchrotron cooling time is ∼1000 yr, cf. the hot gas cooling time ∼10<sup>5</sup> yr, so this requires electron acceleration on this time scale.</p>
</blockquote>

<accretion-discs><sagittarius-a><thermodynamics><synchrotron-radiation>
What does "synchrotron emission from relativistic electrons in equipartition with a ∼1.5 mG magnetic field" mean?
uhoh
# What does "synchrotron emission from relativistic electrons in equipartition with a ∼1.5 mG magnetic field" mean?

(Asked by: username_0 on 2021-03-31)

I've heard of the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem) in the context of thermodynamics.

But I don't understand how either "synchrotron emission" or "relativistic electrons" can be in "equipartition with a ∼1.5 mG magnetic field".

Does it mean that the energy density stored in the magnetic field is equal to (or is equilibrated with) the energy density of the relativistic electrons and/or the synchrotron radiation that they produce?

The phrase is from a block quote in [this extensive & thorough answer to Nature and mechanism of the short-term variability in radio strength of Sgr A\*?](https://astronomy.stackexchange.com/a/43198/7982) citing F. Yusef-Zadeh (2017) [ALMA and VLA observations of emission from the environment of Sgr A\*](https://arxiv.org/abs/1706.02750)

I see "energy density" half-way through the paragraph below, so I think I may be on the right track, but this is over my head:

> 4 DISCUSSION
>
> 4.1 Low Extinction Millimeter Halo
>
> [...]The millimeter halo is coincident with the X-ray emission around Sgr A\*, which is dominated by bremsstrahlung arising from a medium with ne∼150 cm−3 and T∼3×107 K. The bremsstrahlung contribution at 230 GHz, about 0.2 µJy, is negligible. Thermal continuum from dust can also be ruled out because of the halo’s extinction deficit of 0.5 magnitudes at H-band relative to its surroundings. The millimeter emission could, however, be produced by synchrotron emission from relativistic electrons in equipartition with a ∼ 1.5 mG magnetic field. The energy density of each of these components would then be ∼ 10% of the thermal energy density of the hot gas, so this is plausible. The luminosity in the mm is $4 \pi d^2 \nu S\_{\nu}$ ∼1.4×1033 erg s−1, comparable to the X-ray luminosity, LX ∼1×1033 erg s−1, implying that synchrotron cooling is marginally the dominant cooling mechanism for the gas. The synchrotron cooling time is ∼1000 yr, cf. the hot gas cooling time ∼105 yr, so this requires electron acceleration on this time scale.

**Tags:** accretion-discs, sagittarius-a, thermodynamics, synchrotron-radiation

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## 1 answers

### Answer by username_1 on 2021-12-18. Score: 1

It would mean that the kinetic energy density of the electrons was equal to the energy density of the magnetic field.

> **Comments:**
> **username_0**: Oh my goodness I've parsed the sentence differently than you, that's the problem! "X from Y in equipartition with Z"  I read it as "X (from Y) in equipartition with Z" but it's "X, from Y *which is* in equipartition with Z". Yes this solves it. *Thanks!*  

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