Derivation of the formula for longitude of ascending node for a satellite

Question

I've been looking into the document IS-GPS-200H to understand how to calculate satellite location in the ECEF coordinate.

I am having problem understanding the formula to derive $\Omega$, the longitude of the ascending node (LAN) relative to Greenwich at given time $t$:

$$ \Omega = \Omega_0 + \left( \dot{\Omega - w} \right)\times t_k - w \times t_{oe} $$

where: $$ \Omega_0: \text{LAN relative to vernal equinox, at the beginning of the week}\\ \dot{\Omega}: \text{angular velocity for LAN, relative to vernal equinox.}\\ w: \text{angular velocity of earth, relative to vernal equinox.}\\ t_k: t - t_{oe}\\ t_{oe}: \text{ephemeris reference epoch}\\ $$ (and let us denote the beginning of the week as $t_0$ for brevity).

But if I try to work out this from scratch:

1. At $t = t_0$, LAN was $\Omega_0$. But since what we really need is the difference of LAN and longitude of Greenwich, we also need to know $w_0$, the initial longitude of Greenwich at $t = t_0$. $$ \Omega(t = t_0) = \Omega_0 - w_0 $$
2. At the ephemeris reference epoch time $t = t_{oe}$, LAN and the earth both rotate with their respective angular momentum and hence: $$ \Omega(t = toe) = \Omega_0 + w_0 + (\dot{\Omega} - w) \times t_{oe} $$
3. As time varies from $toe$ to $t$, again LAN and the earth both rotate with their own respective angular momentum and hence $$ \Omega(t) = \Omega_0 + w_0 + (\dot{\Omega} - w) \times t_{oe} + (\dot{\Omega} - w) \times t_k $$ which obviously differs from the right formula by $w_0 + \dot{\Omega} \times t_{oe}$.

My question is where am I making mistakes/misunderstanding the eqution? Explain also why we don't need to know $w_0$ or equivalent input, that would be greatly appreciated.

Answer 1

I think you're not too far from understanding the formula. However, there are a couple details you're missing:

1. From what I understand from the document you linked to, $t_0 = t_{oe}$ is the time at the beginning of the week, or reference epoch.
2. When computing an angle from a angular velocity, you usually need to compute it with respect to an initial angle and an initial time: when the Earth and the satellite rotate between $t_0$ and $t$, the angle they rotated of is $\dot{\Omega}(t - t_0)$ or $\dot{\Omega}_E(t - t_0)$.
   The only exception to this rule seems to be the initial Greenwich longitude with respect to the vernal equinox, which seems to be computed from $t = 0$.
3. $w_0$, the initial Greenwich longitude wrt the vernal equinox, is $\dot{\Omega}_E t_{oe}$. From this formula, it seems like the document considers an origin of time $t = 0$ when the longitude of Greenwich wrt to the vernal equinox at this time is 0.

To summarize: the formula would have been more explicit and comprehensible written as: $$ \Omega(t) = \Omega_0 - \dot{\Omega}_E t_{oe} + (\dot{\Omega} - \dot{\Omega}_E)(t - t_{oe}) $$ with:

• $\Omega_0$ = LAN wrt the vernal equinox at $t_{oe}$
• $\dot{\Omega_E}$ = angular velocity of the Earth wrt the vernal equinox
• $-\dot{\Omega}_Et_{oe}$ = Longitude of Greenwich at $t_{oe}$
• $\dot{\Omega}$ = angular velocity of the satellite wrt the vernal equinox
• $(\dot{\Omega} - \dot{\Omega}_E)(t - t_{oe})$ = relative angular displacement between the satellite and Greenwich, between time $t$ and $t_{oe}$

That said, it seems like really bad practice to name $\Omega(t)$ the longitude with respect to Greenwich and $\dot{\Omega}$ the angular velocity with respect to the vernal equinox. This causes quite a bit of confusion, since mathematically speaking, for any linear function $f$, $f(t) = f(t_0) + \dot{f}\times(t - t_0)$