<p>The brightness considered here is the brightness of scattered Lyman $\alpha$ as viewed in the <em>upstream</em> direction, i.e. away from the Sun.</p>

<p>The authors assume that the density of the atoms responsible for scattering the Ly$\alpha$ — the neutral hydrogen atoms — is 1) roughly constant throughout the Solar System and 2) is optically thin, meaning that the radiation field is not attenuated considerably with distance from the Sun. </p>

<p>Since the intensity of a point source falls off as $1/r^2$, the scattering emissivity <em>in a given point</em> should then also fall off roughly as $1/r^2$.</p>

<p>But when viewed from inside the Solar System and in the upstream direction, the brightness is the integral of all scattering events from that point to infinity, and since $\int 1/r^2 \propto 1/r$, you have your quoted brightness.</p>

CC BY-SA 4.0
2018-08-13T15:47:57.793Z
Oh, *scattered* light. Super, Bingo! Nifty! Excellent! yay for math!
uhoh
CC BY-SA 4.0
2018-08-14T08:01:53.447Z
@uhoh I'm a bit reluctant on the assumption of a uniform IPM, though. The _total_ hydrogen density $n_\mathrm{H}$ should decrease away from the Sun, as Kornpob also comments. The _neutral_ hydrogen density $n_\mathrm{HI}$ might decrease less steeply, though, if the ionization fraction also decreases away from the Sun, but I don't think an appreciably large fraction of the IPM is ionized at all, so I'd expect $n_\mathrm{HI}$ to decrease as well. The authors quote Izmodenov (2013) for a uniform $n_\mathrm{HI} = 0.1$ cm$^{-3}$, but a quick browsing through that source didn't really show this…
pela
CC BY-SA 4.0
2018-08-14T12:55:36.273Z
The constant density of neutral H assumption comes from the interstellar medium, not the solar wind.  The solar wind density falls off like 1/r^2, as mentioned, but it highly ionized, and the degree of ionization stays "frozen in" as the temperature drops because the density also drops.  But the ISM is rich with neutrals (it has had time to reach ionization equilibrium), which easily penetrate the interplanetary medium because they are neutral.  The solar wind density at 100 AU is like 0.0006 ions per cc, and the neutrals way less, so the neutrals are from the ISM.  So your answer is right.
Ken G
CC BY-SA 4.0
2018-08-15T07:27:41.573Z
@KenG You're probably right. I was just thinking that dissipation would make atoms loose energy and hence settle down closer to the star. But maybe this is a minor effect.
pela
CC BY-SA 4.0
2018-08-16T04:21:45.530Z
There could be an influence closer to the star, but the distances of interest here are more like 100 AU, and that far away I don't think the star has much effect at all on the ISM neutrals, other than to illuminate them.
Ken G
CC BY-SA 4.0
2018-08-13T13:25:16.717Z
2018-08-13T13:25:16.717Z
pela
<p>The Live Science article <a href="https://www.livescience.com/63297-hydrogen-wall-glowing-interstellar-space.html" rel="nofollow noreferrer">NASA Spotted a Vast, Glowing 'Hydrogen Wall' at the Edge of Our Solar System</a> links to Geophysical Research Letters <a href="https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2018GL078808" rel="nofollow noreferrer">The Lyman‐α Sky Background as Observed by New Horizons</a> <a href="https://doi.org/10.1029/2018GL078808" rel="nofollow noreferrer">https://doi.org/10.1029/2018GL078808</a> the abstract of which is</p>

<blockquote>
  <p>Recent observations of interplanetary medium (IPM) atomic hydrogen Lyman‐α (Lyα) emission in the outer solar system, made with the Alice ultraviolet spectrograph on New Horizons (NH), are presented. The observations include regularly spaced great‐circle scans of the sky and pointed observations near the downstream and upstream flow directions of interstellar H atoms. The NH Alice data agree very well with the much earlier Voyager UVS results, after these are reduced by a factor of 2.4 in brightness, in accordance with recent re‐analyses. <strong>In particular, the falloff of IPM Lyα brightness in the upstream‐looking direction as a function of spacecraft distance from the Sun is well‐matched by an expected 1/r dependence, but with an added constant brightness of ~40 Rayleighs. This additional brightness is a possible signature of the hydrogen wall at the heliopause or of a more distant background</strong>. Ongoing observations are planned at a cadence of roughly twice per year.</p>
</blockquote>

<p>The constant brightness (indicator of "the wall") makes sense to me, a large resolved surface of constant emission intensity (W/m^2) will have a constant brightness, independent of distance.</p>

<p>Conversely, a point source's brightness will obey a $\frac{1}{r^2}$ (<a href="https://en.wikipedia.org/wiki/Inverse-square_law" rel="nofollow noreferrer">inverse square</a>) law.</p>

<p>If absorption were the reason for the fall off, it would be <a href="https://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law" rel="nofollow noreferrer">Beer-Lambert</a> as $e^{-\alpha r}$.</p>

<p>But the abstract describes a 1/r fall off of the interplanetary medium (IPM) atomic hydrogen Lyman‐α (Lyα) emission.</p>

<p><strong>Question:</strong> How does a $\frac{1}{r}$ falloff arise in this case? </p>

CC BY-SA 4.0
2018-08-10T15:47:05.280Z
I don't know how to tag this question so I created `interplanetary-medium`. If someone more knowledgable than I can think of a better existing or new tag, please edit as necessary or make a suggestion. Thanks!
uhoh
CC BY-SA 4.0
2018-08-10T17:18:57.747Z
Since the "edge of the Solar System" is a fuzzy and evolving concept, it's difficult to know whether this is the end of interplanetary and/or the start of interstellar space. So I think you new tag makes more sense compared with the next obvious candidate of `interstellar-medium`
astrosnapper
CC BY-SA 4.0
2018-08-13T15:38:15.760Z
1/r^2 is the density profile of steady flow. You can derive this from the fluid equation. This is normally true for the case of stellar wind and circumstellar medium (i.e., the environment close to a star).

1/r is uniform density profile. (Cannot remember exactly how it is derived). This is the case for interstellar medium (i.e., farther away from the circumstellar medium).

Atomic hydrogen lives interstellar medium. So, its density scales as 1/r.

To connect from density to emission, you just need a scaling relation.
Kornpob Bhirombhakdi
CC BY-SA 4.0
2018-08-13T15:40:23.383Z
@KornpobBhirombhakdi if you put that in an answer, you'll have more space to explain. Right now your comment just demonstrates that you know the answer and you could write one if you wanted to. That's great, but it doesn't help me yet. I would really appreciate it if you write it up as an answer, thanks!
uhoh
CC BY-SA 4.0
2018-08-13T15:43:56.703Z
@uhoh, I have to run. Hopefully someone will give you more details.
Kornpob Bhirombhakdi
CC BY-SA 4.0
2018-08-14T09:50:29.373Z
Note that if this is the only question tagged [interplanetary-medium], and no tag excerpt is written for it, the tag will be deleted after 6 months.
Glorfindel
<interplanetary-medium>
Why would the interplanetary medium atomic hydrogen Lyman‐α emission fall as 1/r rather than1/r^2?
uhoh
# Why would the interplanetary medium atomic hydrogen Lyman‐α emission fall as 1/r rather than1/r^2?

(Asked by: username_0 on 2018-08-10)

The Live Science article [NASA Spotted a Vast, Glowing 'Hydrogen Wall' at the Edge of Our Solar System](https://www.livescience.com/63297-hydrogen-wall-glowing-interstellar-space.html) links to Geophysical Research Letters [The Lyman‐α Sky Background as Observed by New Horizons](https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2018GL078808) <https://doi.org/10.1029/2018GL078808> the abstract of which is

> Recent observations of interplanetary medium (IPM) atomic hydrogen Lyman‐α (Lyα) emission in the outer solar system, made with the Alice ultraviolet spectrograph on New Horizons (NH), are presented. The observations include regularly spaced great‐circle scans of the sky and pointed observations near the downstream and upstream flow directions of interstellar H atoms. The NH Alice data agree very well with the much earlier Voyager UVS results, after these are reduced by a factor of 2.4 in brightness, in accordance with recent re‐analyses. **In particular, the falloff of IPM Lyα brightness in the upstream‐looking direction as a function of spacecraft distance from the Sun is well‐matched by an expected 1/r dependence, but with an added constant brightness of ~40 Rayleighs. This additional brightness is a possible signature of the hydrogen wall at the heliopause or of a more distant background**. Ongoing observations are planned at a cadence of roughly twice per year.

The constant brightness (indicator of "the wall") makes sense to me, a large resolved surface of constant emission intensity (W/m^2) will have a constant brightness, independent of distance.

Conversely, a point source's brightness will obey a $\frac{1}{r^2}$ ([inverse square](https://en.wikipedia.org/wiki/Inverse-square_law)) law.

If absorption were the reason for the fall off, it would be [Beer-Lambert](https://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law) as $e^{-\alpha r}$.

But the abstract describes a 1/r fall off of the interplanetary medium (IPM) atomic hydrogen Lyman‐α (Lyα) emission.

**Question:** How does a $\frac{1}{r}$ falloff arise in this case?

**Tags:** interplanetary-medium
> **Comments:**
> **username_0**: I don't know how to tag this question so I created `interplanetary-medium`. If someone more knowledgable than I can think of a better existing or new tag, please edit as necessary or make a suggestion. Thanks!  
> **username_1**: Since the "edge of the Solar System" is a fuzzy and evolving concept, it's difficult to know whether this is the end of interplanetary and/or the start of interstellar space. So I think you new tag makes more sense compared with the next obvious candidate of `interstellar-medium`  
> **username_2**: 1/r^2 is the density profile of steady flow. You can derive this from the fluid equation. This is normally true for the case of stellar wind and circumstellar medium (i.e., the environment close to a star).  1/r is uniform density profile. (Cannot remember exactly how it is derived). This is the case for interstellar medium (i.e., farther away from the circumstellar medium).  Atomic hydrogen lives interstellar medium. So, its density scales as 1/r.  To connect from density to emission, you just need a scaling relation.  
> **username_0**: @KornpobBhirombhakdi if you put that in an answer, you'll have more space to explain. Right now your comment just demonstrates that you know the answer and you could write one if you wanted to. That's great, but it doesn't help me yet. I would really appreciate it if you write it up as an answer, thanks!  
> **username_2**: @uhoh, I have to run. Hopefully someone will give you more details.  
> **username_3**: Note that if this is the only question tagged [interplanetary-medium], and no tag excerpt is written for it, the tag will be deleted after 6 months.  

---

## 1 answers

### Answer by username_4 on 2018-08-13. Score: 5 | Accepted answer

The brightness considered here is the brightness of scattered Lyman $\alpha$ as viewed in the *upstream* direction, i.e. away from the Sun.

The authors assume that the density of the atoms responsible for scattering the Ly$\alpha$ — the neutral hydrogen atoms — is 1) roughly constant throughout the Solar System and 2) is optically thin, meaning that the radiation field is not attenuated considerably with distance from the Sun.

Since the intensity of a point source falls off as $1/r^2$, the scattering emissivity *in a given point* should then also fall off roughly as $1/r^2$.

But when viewed from inside the Solar System and in the upstream direction, the brightness is the integral of all scattering events from that point to infinity, and since $\int 1/r^2 \propto 1/r$, you have your quoted brightness.

> **Comments:**
> **username_0**: Oh, *scattered* light. Super, Bingo! Nifty! Excellent! yay for math!  
> **username_4**: @uhoh I'm a bit reluctant on the assumption of a uniform IPM, though. The _total_ hydrogen density $n_\mathrm{H}$ should decrease away from the Sun, as Kornpob also comments. The _neutral_ hydrogen density $n_\mathrm{HI}$ might decrease less steeply, though, if the ionization fraction also decreases away from the Sun, but I don't think an appreciably large fraction of the IPM is ionized at all, so I'd expect $n_\mathrm{HI}$ to decrease as well. The authors quote Izmodenov (2013) for a uniform $n_\mathrm{HI} = 0.1$ cm$^{-3}$, but a quick browsing through that source didn't really show this…  
> **username_5**: The constant density of neutral H assumption comes from the interstellar medium, not the solar wind.  The solar wind density falls off like 1/r^2, as mentioned, but it highly ionized, and the degree of ionization stays "frozen in" as the temperature drops because the density also drops.  But the ISM is rich with neutrals (it has had time to reach ionization equilibrium), which easily penetrate the interplanetary medium because they are neutral.  The solar wind density at 100 AU is like 0.0006 ions per cc, and the neutrals way less, so the neutrals are from the ISM.  So your answer is right.  
> **username_4**: @KenG You're probably right. I was just thinking that dissipation would make atoms loose energy and hence settle down closer to the star. But maybe this is a minor effect.  
> **username_5**: There could be an influence closer to the star, but the distances of interest here are more like 100 AU, and that far away I don't think the star has much effect at all on the ISM neutrals, other than to illuminate them.  

---

