Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and

be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple

-ideals

and all the other

-ideals are uniquely factored into a product of those simple ones [17]. Lipman further showed that the predecessor of the smallest simple

-ideal P is either simple or the product of two simple

-ideals. The simple integrally closed ideal P is said to be free for the former and satellite for the later. In this paper we describe the sequence of simple

-ideals when P is satellite of order 3 in terms of the invariant

, where

is the prime divisor associated to P and m = (x, y). Denote

by b and let b = 3k + 1 for k = 0, 1, 2. Let

be the number of nonmaximal simple

-ideals of order i for i = 1, 2, 3. We show that the numbers

= (

,

,

) = (

, 1, 1) and that the rank of P is

= k + 3. We then describe all the

-ideals from m to P as products of those simple

-ideals. In particular, we find the conductor ideal and the

-predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for

. We also find the value semigroup

of a satellite simple valuation ideal P of order 3 in terms of

.