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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.

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I just edited your questions check out this guide which explains how you can enter equations for future reference. –  harogaston Jul 13 at 4:47

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