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40 changes: 26 additions & 14 deletions docs/swapi/alpha-sw.md
Original file line number Diff line number Diff line change
Expand Up @@ -76,21 +76,33 @@ Finally, the `alpha-sw` CDF variables are derived from the fitted parameter vect

| CDF variable | Symbol | Definition |
|---------------------------------------|---------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------|
| `epoch` | $`t_{\text{center}}`$ | center time of the 5-sweep chunk |
| `epoch_delta` | | $`30\thinspace \text{s}`$ (half-width of the 5-sweep chunk) |
| `alpha_sw_density` | $`n^{\alpha}`$ | $`\exp(x^{\alpha}_{0})`$ |
| `alpha_sw_density_uncert` | $`\sigma_{n^{\alpha}}`$ | $`n^{\alpha}\thinspace \sqrt{\Sigma_{x^{\alpha}}[0, 0]}`$ |
| `alpha_sw_temperature` | $`T^{\alpha}`$ | $`\exp(x^{\alpha}_{1})`$ |
| `alpha_sw_temperature_uncert` | $`\sigma_{T^{\alpha}}`$ | $`T^{\alpha}\thinspace \sqrt{\Sigma_{x^{\alpha}}[1, 1]}`$ |
| `alpha_sw_velocity_rtn` | $`\mathbf{v}^{\alpha,\text{SC}}`$ | $`\mathbf{v}_{b}^{\text{SC}} + x^{\alpha}_{2}\thinspace \hat{\mathbf{B}}`$ <sup>[1](#fn-vp)</sup> |
| `alpha_sw_velocity_rtn_covariance` | $`\Sigma_{\mathbf{v}^{\alpha}}`$ | $`\Sigma_{\mathbf{v}^{p}} + \Sigma_{x^{\alpha}}[2,2 ]\thinspace \hat{\mathbf{B}}\hat{\mathbf{B}}^{\top}`$ |
| `alpha_sw_velocity_rtn_uncert` | $`\sigma_{\mathbf{v}^{\alpha}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v}^{\alpha}})}`$ (same for `alpha_sw_velocity_rtn_sun`) |
| `alpha_sw_speed` | $`\lvert\mathbf{v}^{\alpha,\text{SC}}\rvert`$ | $`\lvert\mathbf{v}^{\alpha,\text{SC}}\rvert`$ |
| `alpha_sw_speed_uncert` | $`\sigma_{\lvert\mathbf{v}^{\alpha,\text{SC}}\rvert}`$ | propagated through `uncertainties` from $`\mathbf{v}^{\alpha,\text{SC}}`$ |
| `alpha_sw_velocity_rtn_sun` | $`\mathbf{v}^{\alpha,\text{sun}}`$ | $`\mathbf{v}^{\alpha,\text{SC}} + \mathbf{v}_{\text{sc}}^{\text{sun}}`$ |
| `alpha_sw_speed_sun` | $`\lvert\mathbf{v}^{\alpha,\text{sun}}\rvert`$ | $`\lvert\mathbf{v}^{\alpha,\text{sun}}\rvert`$ |
| `epoch` | $`t_{\text{center}}`$ | center time of the 5-sweep chunk |
| `epoch_delta` | | $`30\thinspace \text{s}`$ (half-width of the 5-sweep chunk) |
| `alpha_sw_density` | $`n^{\alpha}`$ | $`\exp(x^{\alpha}_{0})`$ |
| `alpha_sw_density_uncert` | $`\sigma_{n^{\alpha}}`$ | $`n^{\alpha}\thinspace \sqrt{\Sigma_{x^{\alpha}}[0, 0]}`$ |
| `alpha_sw_temperature` | $`T^{\alpha}`$ | $`\exp(x^{\alpha}_{1})`$ |
| `alpha_sw_temperature_uncert` | $`\sigma_{T^{\alpha}}`$ | $`T^{\alpha}\thinspace \sqrt{\Sigma_{x^{\alpha}}[1, 1]}`$ |
| `alpha_sw_velocity_rtn` | $`\mathbf{v}^{\alpha,\text{SC}}`$ | $`\mathbf{v}_{b}^{\text{SC}} + x^{\alpha}_{2}\thinspace \hat{\mathbf{B}}`$ <sup>[1](#fn-vp)</sup> |
| `alpha_sw_velocity_rtn_covariance` | $`\Sigma_{\mathbf{v}^{\alpha}}`$ | $`\Sigma_{\mathbf{v}^{p}} + \Sigma_{x^{\alpha}}[2,2 ]\thinspace \hat{\mathbf{B}}\hat{\mathbf{B}}^{\top}`$ |
| `alpha_sw_velocity_rtn_uncert` | $`\sigma_{\mathbf{v}^{\alpha}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v}^{\alpha}})}`$ (same for `alpha_sw_velocity_rtn_sun`) |
| `alpha_sw_speed` | $`\lvert\mathbf{v}^{\alpha,\text{SC}}\rvert`$ | $`\lvert\mathbf{v}^{\alpha,\text{SC}}\rvert`$ |
| `alpha_sw_speed_uncert` | $`\sigma_{\lvert\mathbf{v}^{\alpha,\text{SC}}\rvert}`$ | propagated through `uncertainties` from $`\mathbf{v}^{\alpha,\text{SC}}`$ |
| `alpha_sw_velocity_rtn_sun` | $`\mathbf{v}^{\alpha,\text{sun}}`$ | $`\mathbf{v}^{\alpha,\text{SC}} + \mathbf{v}_{\text{sc}}^{\text{sun}}`$ |
| `alpha_sw_velocity_gse` | $`\mathbf{v}^{\alpha,\text{SC}}_{\text{GSE}}`$ | RTN spacecraft-frame velocity rotated to GSE coordinates |
| `alpha_sw_velocity_gse_sun` | $`\mathbf{v}^{\alpha,\text{sun}}_{\text{GSE}}`$ | RTN Sun-frame velocity rotated to GSE coordinates |
| `alpha_sw_velocity_gse_covariance` | $`\Sigma_{\mathbf{v}^{\alpha},\text{GSE}}`$ | $`R_{\text{GSE}}\Sigma_{\mathbf{v}^{\alpha}}R_{\text{GSE}}^{\top}`$ |
| `alpha_sw_velocity_gse_uncert` | $`\sigma_{\mathbf{v}^{\alpha},\text{GSE}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v}^{\alpha},\text{GSE}})}`$ |
| `alpha_sw_velocity_gsm` | $`\mathbf{v}^{\alpha,\text{SC}}_{\text{GSM}}`$ | RTN spacecraft-frame velocity rotated to GSM coordinates |
| `alpha_sw_velocity_gsm_sun` | $`\mathbf{v}^{\alpha,\text{sun}}_{\text{GSM}}`$ | RTN Sun-frame velocity rotated to GSM coordinates |
| `alpha_sw_velocity_gsm_covariance` | $`\Sigma_{\mathbf{v}^{\alpha},\text{GSM}}`$ | $`R_{\text{GSM}}\Sigma_{\mathbf{v}^{\alpha}}R_{\text{GSM}}^{\top}`$ |
| `alpha_sw_velocity_gsm_uncert` | $`\sigma_{\mathbf{v}^{\alpha},\text{GSM}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v}^{\alpha},\text{GSM}})}`$ |
| `alpha_sw_velocity_hae` | $`\mathbf{v}^{\alpha,\text{SC}}_{\text{HAE-J2000}}`$ | RTN spacecraft-frame velocity rotated to HAE-J2000 coordinates |
| `alpha_sw_velocity_hae_sun` | $`\mathbf{v}^{\alpha,\text{sun}}_{\text{HAE-J2000}}`$ | RTN Sun-frame velocity rotated to HAE-J2000 coordinates |
| `alpha_sw_velocity_hae_covariance` | $`\Sigma_{\mathbf{v}^{\alpha},\text{HAE-J2000}}`$ | $`R_{\text{HAE-J2000}}\Sigma_{\mathbf{v}^{\alpha}}R_{\text{HAE-J2000}}^{\top}`$ |
| `alpha_sw_velocity_hae_uncert` | $`\sigma_{\mathbf{v}^{\alpha},\text{HAE-J2000}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v}^{\alpha},\text{HAE-J2000}})}`$ |
| `alpha_sw_speed_sun` | $`\lvert\mathbf{v}^{\alpha,\text{sun}}\rvert`$ | $`\lvert\mathbf{v}^{\alpha,\text{sun}}\rvert`$ |
| `alpha_sw_speed_sun_uncert` | $`\sigma_{\lvert\mathbf{v}^{\alpha,\text{sun}}\rvert}`$ | propagated through `uncertainties` from $`\mathbf{v}^{\alpha,\text{sun}}`$ |
| `swp_flags` | | quality flag bitmask (see `SwapiL3Flags`) |
| `swp_flags` | | quality flag bitmask (see `SwapiL3Flags`) |



Expand Down
13 changes: 12 additions & 1 deletion docs/swapi/proton-sw.md
Original file line number Diff line number Diff line change
Expand Up @@ -198,6 +198,18 @@ The `proton-sw` CDF variables are derived from $`\mathbf{x}`$ and $`\Sigma_\math
| `proton_sw_speed` | $`\lvert\mathbf{v}_{b}^{\text{SC}}\rvert`$ | $`\lvert\mathbf{v}_{b}^{\text{SC}}\rvert`$ |
| `proton_sw_speed_uncert` | $`\sigma_{\lvert\mathbf{v}_{b}^{\text{SC}}\rvert}`$ | propagated through `uncertainties` from $`\mathbf{v}_{b}^{\text{SC}}`$ |
| `proton_sw_velocity_rtn_sun` | $`\mathbf{v}_{b}^{\text{sun}}`$ | $`\mathbf{v}_{b}^{\text{SC}} + \mathbf{v}_{\text{sc}}^{\text{sun}}`$ <sup>[2](#fn-vsc)</sup> |
| `proton_sw_velocity_gse` | $`\mathbf{v}_{b,\text{GSE}}^{\text{SC}}`$ | RTN spacecraft-frame velocity rotated to GSE coordinates |
| `proton_sw_velocity_gse_sun` | $`\mathbf{v}_{b,\text{GSE}}^{\text{sun}}`$ | RTN Sun-frame velocity rotated to GSE coordinates |
| `proton_sw_velocity_gse_covariance` | $`\Sigma_{\mathbf{v},\text{GSE}}`$ | $`R_{\text{GSE}}\Sigma_{\mathbf{v}}R_{\text{GSE}}^{\top}`$ |
| `proton_sw_velocity_gse_uncert` | $`\sigma_{\mathbf{v},\text{GSE}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v},\text{GSE}})}`$ |
| `proton_sw_velocity_gsm` | $`\mathbf{v}_{b,\text{GSM}}^{\text{SC}}`$ | RTN spacecraft-frame velocity rotated to GSM coordinates |
| `proton_sw_velocity_gsm_sun` | $`\mathbf{v}_{b,\text{GSM}}^{\text{sun}}`$ | RTN Sun-frame velocity rotated to GSM coordinates |
| `proton_sw_velocity_gsm_covariance` | $`\Sigma_{\mathbf{v},\text{GSM}}`$ | $`R_{\text{GSM}}\Sigma_{\mathbf{v}}R_{\text{GSM}}^{\top}`$ |
| `proton_sw_velocity_gsm_uncert` | $`\sigma_{\mathbf{v},\text{GSM}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v},\text{GSM}})}`$ |
| `proton_sw_velocity_hae` | $`\mathbf{v}_{b,\text{HAE-J2000}}^{\text{SC}}`$ | RTN spacecraft-frame velocity rotated to HAE-J2000 coordinates |
| `proton_sw_velocity_hae_sun` | $`\mathbf{v}_{b,\text{HAE-J2000}}^{\text{sun}}`$ | RTN Sun-frame velocity rotated to HAE-J2000 coordinates |
| `proton_sw_velocity_hae_covariance` | $`\Sigma_{\mathbf{v},\text{HAE-J2000}}`$ | $`R_{\text{HAE-J2000}}\Sigma_{\mathbf{v}}R_{\text{HAE-J2000}}^{\top}`$ |
| `proton_sw_velocity_hae_uncert` | $`\sigma_{\mathbf{v},\text{HAE-J2000}}`$ | $`\sqrt{\operatorname{diag}(\Sigma_{\mathbf{v},\text{HAE-J2000}})}`$ |
| `proton_sw_speed_sun` | $`v_{\text{sun}}`$ | $`\lvert\mathbf{v}_{b}^{\text{sun}}\rvert`$ |
| `proton_sw_speed_sun_uncert` | $`\sigma_{v_{\text{sun}}}`$ | propagated through `uncertainties` from $`\mathbf{v}_{b}^{\text{sun}}`$ |
| `swp_flags` | | quality flag bitmask (see `SwapiL3Flags`) |
Expand All @@ -207,4 +219,3 @@ The `proton-sw` CDF variables are derived from $`\mathbf{x}`$ and $`\Sigma_\math
<a id="fn-velcov"></a>[1]: The square root of the diagonal of $`\Sigma_{\mathbf{v}}`$ yields the standard error of the $`v_{R}, v_{T}, v_{N}`$ components. The covariance is the same for `proton_sw_velocity_rtn_sun`.

<a id="fn-vsc"></a>[2]: $`\mathbf{v}_{\text{sc}}^{\text{sun}}`$ is the spacecraft velocity in the Sun's inertial frame.

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