Difference between revisions of "Wouthuysen Field effect"

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[[Radiative Processes in Astrophysics|Course Home]]
 +
 
===Reference Materials===
 
===Reference Materials===
 
*  [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Page 32 of Furlanetto et. al, 2009]
 
*  [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Page 32 of Furlanetto et. al, 2009]
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*  [http://www.astro.sunysb.edu/fwalter/AST341/qn.html A Primer on Quantum Numbers and Spectroscopic Notation (Walter, Stony Brook)]
 
*  [http://www.astro.sunysb.edu/fwalter/AST341/qn.html A Primer on Quantum Numbers and Spectroscopic Notation (Walter, Stony Brook)]
 
*  [http://www.columbia.edu/~crg2133/Files/CambridgeIA/Chemistry/Orbitals.pdf  Orbitals (Guetta, Columbia)]
 
*  [http://www.columbia.edu/~crg2133/Files/CambridgeIA/Chemistry/Orbitals.pdf  Orbitals (Guetta, Columbia)]
 +
 +
===Need to Review===
 +
* [[21cm Transition]]
 +
* [[Einstein Coefficients]]
 +
* [[Classical Bohr Atom]]
 +
* [[Atomic and Molecular Quantum Numbers]]
 +
* [[Collisional Excitations]]
 +
* [[Black-Body Radiation]]
 +
 
<latex>
 
<latex>
 
\documentclass{article}
 
\documentclass{article}
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\section*{The Wouthuysen-Field Effect}
 
\section*{The Wouthuysen-Field Effect}
  
The Wouthuysen-Field effect is a coupling of the 21cm hyperfine
+
The Wouthuysen-Field effect is a coupling of the [[21cm Transition|21cm hyperfine
transition to Ly-$\alpha$ radiation.
+
transition]] to Ly-$\alpha$ radiation.
  
 
This is important for possible
 
This is important for possible
Line 33: Line 44:
 
(fine splitting), and the smaller effect of electron-proton spin interaction
 
(fine splitting), and the smaller effect of electron-proton spin interaction
 
(hyperfine splitting). Anti-aligned spins lead to lower energy levels.
 
(hyperfine splitting). Anti-aligned spins lead to lower energy levels.
The splittings of the lowest energy levels are shown in Fig. 1:
+
The splittings of the lowest energy levels are:
  
 
\begin{figure}
 
\begin{figure}
Line 46: Line 57:
 
spin (not to be confused with the $S$ orbital). Then $J=|L+S_{e}|$
 
spin (not to be confused with the $S$ orbital). Then $J=|L+S_{e}|$
 
is the electron total angular momentum, and $F=|L+S_{e}+S_{p}|$ is
 
is the electron total angular momentum, and $F=|L+S_{e}+S_{p}|$ is
the hydrogen total angular momentum %
+
the hydrogen total angular momentum. Note these are vector sums.
\footnote{Note these are vector sums. %
+
}
}.
 
  
 
The splitting between the two $1\ S$ levels is of particular interest.
 
The splitting between the two $1\ S$ levels is of particular interest.
 
It has energy difference $\nu=1.42GHz$, corresponding to wavelength
 
It has energy difference $\nu=1.42GHz$, corresponding to wavelength
 
$\lambda=21cm$. We also define $T_{*}=E_{21cm}/k_{B}=0.068K$. In
 
$\lambda=21cm$. We also define $T_{*}=E_{21cm}/k_{B}=0.068K$. In
the $21cm$ regime, the Rayleigh-Jeans limit holds and we can define
+
the $21cm$ regime, the [[Black-Body Radiation|Rayleigh-Jeans limit]] holds and we can define
 
the ``brightness temperature''  
 
the ``brightness temperature''  
 
\[
 
\[
Line 71: Line 81:
 
redshift and hence the actual observed brightness temperature is  
 
redshift and hence the actual observed brightness temperature is  
 
\[
 
\[
T_{b}=T'_{b}/(1+z).
+
T_{b}=T^\prime_{b}/(1+z).
 
\]
 
\]
  
Line 83: Line 93:
 
During the dark ages till the end of reionization, several processes
 
During the dark ages till the end of reionization, several processes
 
control the relative abundances. Among them are direct radiative transitions
 
control the relative abundances. Among them are direct radiative transitions
(Einstein coefficients) %
+
([[Einstein Coefficients]]) %
 
\footnote{Direct transitions are disfavored by selection rules, as we shall
 
\footnote{Direct transitions are disfavored by selection rules, as we shall
 
explain later. %
 
explain later. %
}, collisional excitation $C_{01}$, $C_{10}$ and the Wouthuysen-Field
+
}, [[Collisional Excitations|collisional excitation]] $C_{01}$, $C_{10}$ and the Wouthuysen-Field
 
(WF) Effect $W_{01}$ and $W_{10}$. The WF effect will be explained
 
(WF) Effect $W_{01}$ and $W_{10}$. The WF effect will be explained
 
in detail. For now let's just note that the overall statistical balance
 
in detail. For now let's just note that the overall statistical balance
Line 94: Line 104:
 
\end{equation}
 
\end{equation}
 
where $A_{10}=2.869\times10^{-15}\mbox{s}^{-1}$ and $B$ are the
 
where $A_{10}=2.869\times10^{-15}\mbox{s}^{-1}$ and $B$ are the
Einstein coefficients of the 21cm transition. Recall that in thermodynamic
+
[[Einstein Coefficients]] of the 21cm transition. Recall that in thermodynamic
 
equilibrium, we had  
 
equilibrium, we had  
 
\[
 
\[
 
B_{01}=\frac{g_{1}}{g_{0}}B_{10}=\frac{3c^{2}}{2h\nu^{3}}A_{10}
 
B_{01}=\frac{g_{1}}{g_{0}}B_{10}=\frac{3c^{2}}{2h\nu^{3}}A_{10}
 
\]
 
\]
In the Rayleigh-Jeans limit, radiative coefficients satisfy
+
In the [[Black-Body Radiation|Rayleigh-Jeans limit]], radiative coefficients satisfy
 
\begin{equation}
 
\begin{equation}
 
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\
 
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\
 
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},
 
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},
 
\end{aligned}
 
\end{aligned}
\label{eq:BI}
 
 
\end{equation}
 
\end{equation}
 
Here $T_{\gamma}=2.74(1+z)\mbox{ K }$ is the temperature of the CMB.  
 
Here $T_{\gamma}=2.74(1+z)\mbox{ K }$ is the temperature of the CMB.  
Line 160: Line 169:
 
theorem which applies to ``irreducible tensor operators'' in general,
 
theorem which applies to ``irreducible tensor operators'' in general,
 
and can generate selection rules for any such operator. The selection
 
and can generate selection rules for any such operator. The selection
rule goes like this: a transition from $j$ to $j'$ is only allowed if
+
rule goes like this: a transition from $j$ to $j^/prime$ is only allowed if
$j'$ is in the range $|j-k|$ to $j+k$, where $k$ is the ``order'' of
+
$j^/prime$ is in the range $|j-k|$ to $j+k$, where $k$ is the ``order'' of
 
the tensor operator. For dipole transitions, $k = 1$: the dipole
 
the tensor operator. For dipole transitions, $k = 1$: the dipole
 
operator is an ``order 1 irreducible tensor operator'', so the rule
 
operator is an ``order 1 irreducible tensor operator'', so the rule
becomes $j' = |j-1|, ..., j+1$. (Lowercase $j$ is a totally generic angular
+
becomes $j^/prime = |j-1|, ..., j+1$. (Lowercase $j$ is a totally generic angular
 
momentum quantum number, not the same as $J$.)
 
momentum quantum number, not the same as $J$.)
  
Line 231: Line 240:
 
Thus the matrix element of the vector can be written as  
 
Thus the matrix element of the vector can be written as  
 
\[
 
\[
\begin{aligned}M & =\langle n'l'm'|A^{\mu}|nlm\rangle,\\
+
\begin{aligned}M & =\langle n^/prime l^/prime m^/prime|A^{\mu}|nlm\rangle,\\
  & =\langle n'l'm'|P^{\dagger}PA^{\mu}P^{\dagger}P|nlm\rangle,\\
+
  & =\langle n^/prime l^/prime m^/prime|P^{\dagger}PA^{\mu}P^{\dagger}P|nlm\rangle,\\
  & =(-1)^{l-l'+1}\langle n'l'm'|A^{\mu}|nlm\rangle.
+
  & =(-1)^{l-l^/prime+1}\langle n^/prime l^/prime m^/prime|A^{\mu}|nlm\rangle.
 
\end{aligned}
 
\end{aligned}
 
\]
 
\]
 
Since a matrix element is a scalar and has even parity, we see that
 
Since a matrix element is a scalar and has even parity, we see that
 
the end state must have different parity from the initial state. In
 
the end state must have different parity from the initial state. In
other words, $l'-l$ must be odd!
+
other words, $l^/prime-l$ must be odd!
  
 
Furthermore, since the photon carries only one unit of angular momentum,
 
Furthermore, since the photon carries only one unit of angular momentum,
Line 262: Line 271:
 
The matrix element can be written as  
 
The matrix element can be written as  
 
\[
 
\[
\begin{aligned}\langle n'l'm'|d|nlm\rangle & \propto\int_{0}^{\pi}P_{l'}^{m'}P_{l}^{m}\cos\theta\sin\theta d\theta\times\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(m-m')\phi}d\phi.\end{aligned}
+
\begin{aligned}\langle n^/prime l^/prime m^/prime|d|nlm\rangle & \propto\int_{0}^{\pi}P_{l^/prime}^{m^/prime}P_{l}^{m}\cos\theta\sin\theta d\theta\times\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(m-m^/prime)\phi}d\phi.\end{aligned}
 
\]
 
\]
 
From properties of associated Legendre polynomials, we can decompose
 
From properties of associated Legendre polynomials, we can decompose
Line 271: Line 280:
 
orthogonality rules
 
orthogonality rules
 
\[
 
\[
\int_{-1}^{1}P_{l'}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l'l},
+
\int_{-1}^{1}P_{l^/prime}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l^/prime l},
 
\]
 
\]
 
the above matrix element vanishes unless
 
the above matrix element vanishes unless
 
\[
 
\[
 
\begin{cases}
 
\begin{cases}
m' & =m,\\
+
m^/prime & =m,\\
l' & =l\pm1.
+
l^/prime & =l\pm1.
 
\end{cases}
 
\end{cases}
 
\]
 
\]
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\subsubsection*{Coupling coefficient}
 
\subsubsection*{Coupling coefficient}
 
+
We would like to find the coupling coefficient $x_W$ defined at the end of the Cosmological Context subsection.
 
The allowed transitions between the lowest levels are shown in Fig.
 
The allowed transitions between the lowest levels are shown in Fig.
 
2. The transitions relevant to the WF effect are traced with solid
 
2. The transitions relevant to the WF effect are traced with solid
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\label{eq:W01}
 
\label{eq:W01}
 
\end{equation}
 
\end{equation}
In the second and fourth lines I have used Eq. \eqref{eq:BI}.
+
In the second and fourth lines I have used the relation from above:
 +
\begin{equation}
 +
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\
 +
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},
 +
\end{aligned}
 +
\end{equation}  
  
 
We would like to relate this rate to the total Ly$\alpha$ scattering
 
We would like to relate this rate to the total Ly$\alpha$ scattering
Line 374: Line 388:
  
 
Since the dipole operator commutes with spin, the transition of $|nJFm_{F}\rangle$
 
Since the dipole operator commutes with spin, the transition of $|nJFm_{F}\rangle$
to $|n'J'F'm'_{F}\rangle$ is independent of $F$ and $m_{F}$. This
+
to $|n^/prime J^/prime F^/prime m^/prime_{F}\rangle$ is independent of $F$ and $m_{F}$. This
 
means that the emission intensity of transition from $|nJF\rangle$,
 
means that the emission intensity of transition from $|nJF\rangle$,
summed over all $m_{F}$, to $|n'J'\rangle$, summed over $m'_{F}$
+
summed over all $m_{F}$, to $|n^/prime J^/prime\rangle$, summed over $m^/prime_{F}$
and $F'$, is the intensity of transition from a particular $|nJFm_{F}\rangle$,
+
and $F^/prime$, is the intensity of transition from a particular $|nJFm_{F}\rangle$,
 
times $2F+1$, the degeneracy of the initial state. This is called
 
times $2F+1$, the degeneracy of the initial state. This is called
 
the ``sum rule'' of the transition.  
 
the ``sum rule'' of the transition.  
Line 399: Line 413:
 
\]
 
\]
 
Let $I_{tot}=I_{51}+I_{40}+I_{41}+I_{31}+I_{30}+I_{21}$ be the total
 
Let $I_{tot}=I_{51}+I_{40}+I_{41}+I_{31}+I_{30}+I_{21}$ be the total
intensity of de-excitation. Then from Eq. \eqref{eq:int} we get  
+
intensity of de-excitation. Then from the relations on $I$ above we get  
 
\[
 
\[
 
\begin{aligned}I_{30}=I_{41} & =\frac{1}{12}I_{tot},\\
 
\begin{aligned}I_{30}=I_{41} & =\frac{1}{12}I_{tot},\\
Line 419: Line 433:
 
\end{aligned}
 
\end{aligned}
 
\]
 
\]
Thus finally $P_{10}=4P_{\alpha}/27$ and the coupling coefficient
+
Thus finally $P_{10}=4P_{\alpha}/27$ and the coupling coefficient becomes  
in Eq. \eqref{eq:coeff} becomes  
 
 
\[
 
\[
 
\boxed{x_{W}=\frac{4P_{\alpha}}{27A_{10}}\frac{T_{*}}{T_{\gamma}}}.
 
\boxed{x_{W}=\frac{4P_{\alpha}}{27A_{10}}\frac{T_{*}}{T_{\gamma}}}.

Latest revision as of 10:53, 20 August 2021

Course Home

Reference Materials[edit]

Need to Review[edit]

The Wouthuysen-Field Effect

The Wouthuysen-Field effect is a coupling of the 21cm hyperfine transition to Ly- radiation.

This is important for possible high-redshift observations of the 21cm line, during the epoch of reionization. The 21cm hyperfine transition is forbidden by normal dipole selection rules, but transitions from one hyperfine state, up to the state, and back down to the other hyperfine state are not forbidden. So, if there is sufficient Ly- radiation to cause this intermediate transition, it will dominate over the direct (forbidden) hyperfine transition. Cosmologically speaking, this happens towards the end of the “Dark Ages” as reionization begins, and the WF effect remains the dominant effect until reionization is complete.

Fine structure of hydrogen

Energy levels in hydrogen atoms are split due to spin orbit interaction (fine splitting), and the smaller effect of electron-proton spin interaction (hyperfine splitting). Anti-aligned spins lead to lower energy levels. The splittings of the lowest energy levels are:

WFfig1.jpg
Energy level diagram with spectroscopic notation

Here we have used spectroscopic notation , where is the principal quantum number, are the electron orbital angular momentum. () shall denote the electron (proton) spin (not to be confused with the orbital). Then is the electron total angular momentum, and is the hydrogen total angular momentum. Note these are vector sums.

The splitting between the two levels is of particular interest. It has energy difference , corresponding to wavelength . We also define . In the regime, the Rayleigh-Jeans limit holds and we can define the “brightness temperature”

only serves to redefine , and should be distinguished from the ambient temperature of the CMB .

Cosmological context

After recombination , the photons in the cosmic fluid no longer has free charges to interact with and thus free stream through the “dark ages” until the Epoch of Reionization, believed to be around . Studying the 21cm absorption of neutral hydrogen can thus potentially probe the universe during the dark ages and epoch of reionization. The frequency of 21cm undergoes cosmological redshift and hence the actual observed brightness temperature is

Denoting the lower level by 0, the higher by 1, one can define the “spin temperature ” to characterize the relative abundance of these two states:

During the dark ages till the end of reionization, several processes control the relative abundances. Among them are direct radiative transitions (Einstein Coefficients) [1], collisional excitation , and the Wouthuysen-Field (WF) Effect and . The WF effect will be explained in detail. For now let’s just note that the overall statistical balance gives

where and are the Einstein Coefficients of the 21cm transition. Recall that in thermodynamic equilibrium, we had

In the Rayleigh-Jeans limit, radiative coefficients satisfy

Here is the temperature of the CMB.

Remember that for collisional excitations of the hyperfine transition, in thermal equilibrium, the gas temperature is coupled directly to the spin temperature. In other words, collisions are the dominant effect that sets the hyperfine level population. This means the transition rates and are given by:

Similarly, WF effect rates are given by the “color temperature” :

Plugging the above results into the balance equation eq:balance, we have the relation of the temperatures:

where

represents the relative rates. It remains to determine these coefficients.

The rest of this article shall concentrate on the WF effect, and in particular the determination of The WF effect involves absorption of an Lyman- photon from the state and subsequent decay into the state. The CMB temperature during the dark ages have temperatures , and is much smaller than the Lyman- transition temperature of , and thus we expect WF effect to be most important during the epoch of reionization, when the first stars provide the abundance of photons. To study the WF effect, we first look at the rules for allowed transitions.

Parity and Selection Rules

Now we need to look at the selection rules for dipole transitions. First of all, there is the selection rule that is forbidden by parity, a symmetry of electromagnetic interactions, so only transitions between and will matter. (In other words, nevermind about the states we wrote down above.) This is the selection rule that forbids the 21cm transition, which is why the WF effect can dominate over the direct 21cm transition so long as enough Ly- photons are around.

Rotational symmetry is where the rest of our selection rules come from. There is a fancy thing in quantum called the Wigner-Eckart theorem which applies to “irreducible tensor operators” in general, and can generate selection rules for any such operator. The selection rule goes like this: a transition from to is only allowed if is in the range to , where is the “order” of the tensor operator. For dipole transitions, : the dipole operator is an “order 1 irreducible tensor operator”, so the rule becomes . (Lowercase is a totally generic angular momentum quantum number, not the same as .)

To put illustrate all of these in simpler language, a photon is a vector and carries one unit of angular momentum. This means that a single photon must be circularly polarized (left or right) [2]. Since the vector does not has any spin dependence, it commutes with the electron spin operator:

Here is the vector sum of the electron and proton spins.

This implies the first dipole selection rule:

The spin along given directions (), however, are not “good” quantum numbers, and can change.

To consider orbital angular momentum, recall the parity operator . It reverses all physical space directions:

If we fix the phase such that , we have parity as a Hermitian and hence observable:

Parity thus acts on other operators via conjugation. For example

More generally, it is clear that quantities such as vector, pseudo-scalar change sign under parity while a pseudo-vector (such as angular momentum or magnetic field) or a true scalar do not change sign. For example, parity commutes with spin , while it anti-commutes with momentum . This means that while a spin state is a good quantum number when paired with parity, a momentum state is not.

So how do states in a hydrogen atom behave under parity? To answer this question recall that such a state is proportional to the spherical harmonics:

Under parity, , , and since we have

The parity of a state does not depend on , as expected since an intrinsic property such as parity should not depend on the orientation of the system.

Thus the matrix element of the vector can be written as

Since a matrix element is a scalar and has even parity, we see that the end state must have different parity from the initial state. In other words, must be odd!

Furthermore, since the photon carries only one unit of angular momentum, any angular momentum change must be of order unity or less:

Thus we have the rule

To see this more explicitly, one can also write out the full expression

Since a dipole interaction, along a direction called has the form

The matrix element can be written as

From properties of associated Legendre polynomials, we can decompose

for some constants and . And since they also satisfy the orthogonality rules

the above matrix element vanishes unless

In particular, the angular momentum along the dipole direction is unchanged. This is expected since the spin of the photon is pointed perpendicular to the field (a transverse wave).

A difference between and is in place when there are more than one electron in the atom. We have seen that a single electron contributes a factor of when acted on by parity. In the multi-electron case, this generalizes to

Since in general the total orbital angular momentum , two states with the same can have different parity, and so one cannot conclude that is not allowed. One can conclude, however, that transitions are not allowed. This is because the orbital has complete spherical symmetry, a symmetry broken by the vector photon, and thus an orbital cannot absorb a photon and stay an orbital. Similarly, considering L-S coupling, we have corresponding selection rules of forbidden or transitions. Physically, these last two rules amounts to the combined effect of angular momentum conservation, and the first rule, that For example, the transition from to is allowed, even though , because in this transition the photon brought , electron flips its spin to keep , and proton also flips its spin, to preserve and , in the process is flipped. The transition to on the other hand, is forbidden, because as , the two conditions and cannot be simultaneously satisfied.

In summary, we have the following selection rules for dipole transitions in a neutral hydrogen atom

The WF effect

Coupling coefficient

We would like to find the coupling coefficient defined at the end of the Cosmological Context subsection. The allowed transitions between the lowest levels are shown in Fig. 2. The transitions relevant to the WF effect are traced with solid lines. Since we have denoted the states by 0 and 1, we shall by convenience denote the 2 states by 2,3,4,5, in order of increasing energy.

WFfig2.jpg
Energy level diagram with allowed transitions
WFfig3a.jpg
Energy level diagram of WF effect

Evidently, the (de-)excitation rate due to WF effect is given by:

In the second and fourth lines I have used the relation from above:

We would like to relate this rate to the total Ly scattering rate

where is the absorption cross section and , with the oscillator strength. The line can be assumed to be a Voigt profile. Thermal broadening leads to Doppler width

where is the Lyman line center frequency.

Since the dipole operator commutes with spin, the transition of to is independent of and . This means that the emission intensity of transition from , summed over all , to , summed over and , is the intensity of transition from a particular , times , the degeneracy of the initial state. This is called the “sum rule” of the transition.

The sum rules, together with the selected transitions in Fig. 2, immediately lead to

The second line follow from the transition into excited states.

Next we shall neglect radiative transfer effect and make the assumption that background intensities and temperatures are constant across all the hyperfine lines, and are given by the CMB values. Under this assumption

Let be the total intensity of de-excitation. Then from the relations on above we get

Furthermore, the intensities are related to the Einstein coefficients by

where and is the total degeneracy of level.

Thus we have

Thus finally and the coupling coefficient becomes

Color temperature and corrections

The effect of Ly absorption on spin temperature also depends on the color temperature . For an environment that’s optically thick, such as the high-redshift IGM, we may take

One commonly considered correction is due to energy loss in spin-exchange collisions. In these collisions, the electron spins of the two colliding hydrogen atoms exchange, while the total spin remains unchanged. This correction to the color temperature is worked out in for example [, section 2.3.3. We shall here just quote the result:

where the spin exchange temperature

To use Eq. eq:colort, the spin temperature must be determined iteratively. The iteration is shown to converge quickly.

Another commonly considered correction are radiative transfer effects. Previously we assumed that the background intensity is constant and given by the CMB intensity. The Ly absorption would decrease the background intensity and hence the estimated scattering rate. The effect is more important for lower kinetic temperature. This leads to a correction to the WF effect coupling coefficient such that

where

Neglecting spin exchange, the suppression factor is given by

where the Gunn-Peterson optical depth can be written for overdensity as roughly:

[1] Direct transitions are disfavored by selection rules, as we shall explain later.

[2] In general, a tensor of rank carries units of angular momentum, and has possible spin states. Due to the lack of mass of a photon however, the electromagnetic wave is transverse in all reference frames (easily seen with Maxwell’s equations), and thus only has two polarization states, with angular momentum (anti-)parallel to the direction of motion.