<p>Thermal inertia for asteroids is defined by <a href="https://arxiv.org/abs/0704.1915" rel="nofollow noreferrer">Delbo et al. 2007</a> as
<span class="math-container">$\Gamma = \sqrt{\rho \kappa c}$</span>, where <span class="math-container">$\rho$</span> is the density, <span class="math-container">$\kappa$</span> is the thermal conductivity, and <span class="math-container">$c$</span> is the specific heat capacity.  A more intuitive way of thinking about thermal inertia is the ability of surface material to retain and re-radiate solar heat.  After dark, a sandy beach quickly returns to cool night-time atmospheric temperatures (low thermal inertia), but thick stone canyon walls can continue to exude heat late into the night (high thermal inertia).</p>
<p>Consider two hemispheres of an asteroid, separated by a plane perpendicular to the instantaneous orbital velocity.  There is a &quot;forward&quot; hemisphere pointed to the forward direction of the asteroid's orbit, and a &quot;back&quot; hemisphere pointed the opposite direction.</p>
<p>The forward hemisphere is releasing photons with momentum.  This provides a backwards &quot;push&quot; on the asteroid, a force equal and opposite to the forward component of the sum of the momentum vectors of the photons.  Similarly, the back hemisphere gets a forward &quot;push&quot; from the photons emitted from its surface.  The differences in these values give the magnitude of the Yarkovsky force.  For prograde rotating asteroids, this pushes them into bigger orbits.  For retrograde orbits, the effect spirals them into closer orbits.</p>
<p>The Yarkovsky effect is driven by differences in temperature on the surface of an asteroid. So the Yarkovsky effect won't even exist if the temperature profile of the two hemispheres are symmetrical.  This would be for a tidally locked body, or for a body with spin axis pointed towards the Sun, or for the theoretical case of zero thermal inertia. In the case of high thermal inertia, with a rapidly rotating body, there would be little differences between the hemisphere surface temperatures, so the Yarkovsky effect would be minimized.</p>
<p><strong>Answer:</strong> Let's assume an atmosphere-less, prograde, spherical, uniform body with rotation axis perpendicular to the orbital plane (zero obliquity), with low eccentricity.</p>
<p>Maximum Yarkovsky effect will be produced when:</p>
<ol>
<li>albedo is low (most of the solar energy is absorbed rather than reflected)</li>
<li>emissivity is high (solar energy is easily absorbed and re-radiated)</li>
<li>thermal inertia is such that it maximizes the temperature differences between the &quot;forward&quot; and &quot;back&quot; hemispheres.  This means retaining most of the solar energy through the evening hours (through the &quot;back&quot; hemisphere), but having mostly cooled by morning (through the &quot;forward&quot; hemisphere).  To maximize the Yarkovsky effect, thermal inertia will low for fast rotating bodies, and high for slow rotating bodies.</li>
</ol>

CC BY-SA 4.0
2022-08-02T22:45:24.953Z
2022-08-02T22:45:24.953Z
Connor Garcia
<p>The video below summarizes current work at asteroid <a href="https://en.wikipedia.org/wiki/101955_Bennu" rel="noreferrer">101955 Bennu</a> by OSIRIS-REx to characterize its properties to provide better estimates of the <a href="https://en.wikipedia.org/wiki/Yarkovsky_effect" rel="noreferrer">Yarkovsky effect</a><sup>1</sup>'s impact on its trajectory for an approach to Earth in the 22nd century.</p>
<blockquote>
<p>OSIRIS-REx greatly improved our knowledge of Bennu’s position, density, <strong>thermal inertia</strong>, and other properties that can influence how its orbit will evolve over time. The new data allowed scientists to significantly reduce uncertainties in Bennu’s predicted orbit, ruling out a number of keyholes for the 2135 flyby, and eliminating several future impact scenarios.</p>
</blockquote>
<p><strong>Question:</strong> How do things like thermal inertia, visible light albedo, thermal infrared emissivity and rotation rate interact to produce Yarkovsky? For a given scenario is there a certain inertia that produces &quot;the maximum Yarkovsky&quot;?</p>
<hr />
<p>Related:</p>
<ul>
<li><a href="https://astronomy.stackexchange.com/q/45179/7982">How can a 1-pixel image of a rotating asteroid be used to measure its thermal inertia?</a></li>
<li><a href="https://space.stackexchange.com/q/41550/12102">Do all dangerous asteroids first pass through keyholes?</a></li>
<li><a href="https://space.stackexchange.com/q/13045/12102">Understanding gravitational keyhole analysis for Near-Earth Objects</a></li>
<li>Wikipedia's <a href="https://en.wikipedia.org/wiki/Gravitational_keyhole" rel="noreferrer">Gravitational keyhole</a></li>
</ul>
<hr />
<p><sup>1</sup>not to be confused with the <a href="https://en.wikipedia.org/wiki/Yarkovsky%E2%80%93O%27Keefe%E2%80%93Radzievskii%E2%80%93Paddack_effect" rel="noreferrer">Yarkovsky–O'Keefe–Radzievskii–Paddack (YORP) effect</a>. For more on that see <a href="https://astronomy.stackexchange.com/q/40490/7982">What is the YORP effect exactly? Is it just the non-central component of the Yarkovsky effect?</a></p>
<hr />
<p><div class="youtube-embed"><div>
                <iframe width="640px" height="395px" src="https://www.youtube.com/embed/Cb9IL8AqrGA?start=0"></iframe>
            </div></div></p>

CC BY-SA 4.0
2021-08-12T05:25:35.247Z
@ConnorGarcia thanks! This question is [outstanding in it's field](https://i.sstatic.net/zFaxN.png). https://www.youtube.com/watch?v=C7rOQ0rofYA
uhoh
CC BY-SA 4.0
2021-08-12T12:17:23.240Z
@DavidHammen Thanks for that, how does it look now?
uhoh
CC BY-SA 4.0
2021-08-12T12:46:10.280Z
@DavidHammen I just simply missed those, it's been a long day. I mean to ask about vanilla Yarkovsky; rotation of heated and now cooling areas such that they produce some thrust prograde or retrograde. I don't mean to go beyond that in the question. I just grabbed the wrong "Y" effect when drafting the post.
uhoh
CC BY-SA 4.0
2021-08-12T12:50:59.180Z
Thanks for the clarifications! I'm going to delete my previous comments as you have very adequately addressed them. I'll add part of my last comment as it might steer someone more knowledgeable toward an answer: A tidally locked body will have a minimal Yarkovsky effect, but so will an object that is rotating very quickly. I suspect there's a goldilocks zone with regard to maximizing the Yarkovsky effect regarding rotation rate, thermal inertia, and black body characteristics (or lack thereof).
David Hammen
<asteroids><orbital-mechanics><near-earth-object>
Is there a certain thermal inertia for maximum Yarkovsky?
uhoh
# Is there a certain thermal inertia for maximum Yarkovsky?

(Asked by: username_0 on 2021-08-12)

The video below summarizes current work at asteroid [101955 Bennu](https://en.wikipedia.org/wiki/101955_Bennu) by OSIRIS-REx to characterize its properties to provide better estimates of the [Yarkovsky effect](https://en.wikipedia.org/wiki/Yarkovsky_effect)1's impact on its trajectory for an approach to Earth in the 22nd century.

> OSIRIS-REx greatly improved our knowledge of Bennu’s position, density, **thermal inertia**, and other properties that can influence how its orbit will evolve over time. The new data allowed scientists to significantly reduce uncertainties in Bennu’s predicted orbit, ruling out a number of keyholes for the 2135 flyby, and eliminating several future impact scenarios.

**Question:** How do things like thermal inertia, visible light albedo, thermal infrared emissivity and rotation rate interact to produce Yarkovsky? For a given scenario is there a certain inertia that produces "the maximum Yarkovsky"?

---

Related:

* [How can a 1-pixel image of a rotating asteroid be used to measure its thermal inertia?](https://astronomy.stackexchange.com/q/45179/7982)
* [Do all dangerous asteroids first pass through keyholes?](https://space.stackexchange.com/q/41550/12102)
* [Understanding gravitational keyhole analysis for Near-Earth Objects](https://space.stackexchange.com/q/13045/12102)
* Wikipedia's [Gravitational keyhole](https://en.wikipedia.org/wiki/Gravitational_keyhole)

---

1not to be confused with the [Yarkovsky–O'Keefe–Radzievskii–Paddack (YORP) effect](https://en.wikipedia.org/wiki/Yarkovsky%E2%80%93O%27Keefe%E2%80%93Radzievskii%E2%80%93Paddack_effect). For more on that see [What is the YORP effect exactly? Is it just the non-central component of the Yarkovsky effect?](https://astronomy.stackexchange.com/q/40490/7982)

---

**Tags:** asteroids, orbital-mechanics, near-earth-object

> **Comments:**
> **username_0**: @username_2 thanks! This question is [outstanding in it's field](https://i.sstatic.net/zFaxN.png). https://www.youtube.com/watch?v=C7rOQ0rofYA  
> **username_0**: @username_1 Thanks for that, how does it look now?  
> **username_0**: @username_1 I just simply missed those, it's been a long day. I mean to ask about vanilla Yarkovsky; rotation of heated and now cooling areas such that they produce some thrust prograde or retrograde. I don't mean to go beyond that in the question. I just grabbed the wrong "Y" effect when drafting the post.  
> **username_1**: Thanks for the clarifications! I'm going to delete my previous comments as you have very adequately addressed them. I'll add part of my last comment as it might steer someone more knowledgeable toward an answer: A tidally locked body will have a minimal Yarkovsky effect, but so will an object that is rotating very quickly. I suspect there's a goldilocks zone with regard to maximizing the Yarkovsky effect regarding rotation rate, thermal inertia, and black body characteristics (or lack thereof).  

---

## 1 answers

### Answer by username_2 on 2022-08-02. Score: 1

Thermal inertia for asteroids is defined by [Delbo et al. 2007](https://arxiv.org/abs/0704.1915) as
$\Gamma = \sqrt{\rho \kappa c}$, where $\rho$ is the density, $\kappa$ is the thermal conductivity, and $c$ is the specific heat capacity. A more intuitive way of thinking about thermal inertia is the ability of surface material to retain and re-radiate solar heat. After dark, a sandy beach quickly returns to cool night-time atmospheric temperatures (low thermal inertia), but thick stone canyon walls can continue to exude heat late into the night (high thermal inertia).

Consider two hemispheres of an asteroid, separated by a plane perpendicular to the instantaneous orbital velocity. There is a "forward" hemisphere pointed to the forward direction of the asteroid's orbit, and a "back" hemisphere pointed the opposite direction.

The forward hemisphere is releasing photons with momentum. This provides a backwards "push" on the asteroid, a force equal and opposite to the forward component of the sum of the momentum vectors of the photons. Similarly, the back hemisphere gets a forward "push" from the photons emitted from its surface. The differences in these values give the magnitude of the Yarkovsky force. For prograde rotating asteroids, this pushes them into bigger orbits. For retrograde orbits, the effect spirals them into closer orbits.

The Yarkovsky effect is driven by differences in temperature on the surface of an asteroid. So the Yarkovsky effect won't even exist if the temperature profile of the two hemispheres are symmetrical. This would be for a tidally locked body, or for a body with spin axis pointed towards the Sun, or for the theoretical case of zero thermal inertia. In the case of high thermal inertia, with a rapidly rotating body, there would be little differences between the hemisphere surface temperatures, so the Yarkovsky effect would be minimized.

**Answer:** Let's assume an atmosphere-less, prograde, spherical, uniform body with rotation axis perpendicular to the orbital plane (zero obliquity), with low eccentricity.

Maximum Yarkovsky effect will be produced when:

1. albedo is low (most of the solar energy is absorbed rather than reflected)
2. emissivity is high (solar energy is easily absorbed and re-radiated)
3. thermal inertia is such that it maximizes the temperature differences between the "forward" and "back" hemispheres. This means retaining most of the solar energy through the evening hours (through the "back" hemisphere), but having mostly cooled by morning (through the "forward" hemisphere). To maximize the Yarkovsky effect, thermal inertia will low for fast rotating bodies, and high for slow rotating bodies.


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