Rule of cool
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I think the modified J79 F-4J would work from the bow cats. The question is rather, how well - safety margins, weight limitations etc. But that was a problem for the F-4K, too.
How much of a problem in comparison is difficult to say as it's based on 60 year old bits of data and we lack details - eg, the minimum launch speed from the manual is based on how many ft of sink? But I think it is quite plausible that the advantage of the Spey, if any, was 5 kts at best.
I do not think that was critical. Also to consider, the F-4K was 500-1000 lbs heavier, so a part of the advantage would be eaten up by that.
Given the promise of the Spey, it's easy to see why they chose it, the issue is with under-delivering on that promise and the cost. As you say, even with underperformance the Spey still seems to have a few knots advantage over the J79, so the issue becomes one of value for money.
Here's a tangential question, if the Spey did live up to its promise and deliver the full ~12kts among other things, would it have been worth the money?
EDIT: Post #473
Other costs I've found are that Spey Phantom development cost was estimated at 80-90m pounds in 1965, and the unit cost was 50% more than US versions. Given the 1969 US F4J cost about $3m that would make the F4K/M about $4.5m, or 1.875m pounds.
Hansard Friday 14 April 1967 says the unit cost of the Spey Phantom was 'about 1 1/4m pounds'. With the $2.80 exchange rate and a US cost of maybe a touch under $3m for a 1967 F4J and a touch under $2m for a 1966 F4B this might be pretty close to the mark. Devaluation occurred in November 1967, which would have added 14% to that price, and inflation started to pick up in the late 60s.
The total programme cost for the 170 Spey Phantoms for the RN and RAF by 1969 was about 500 million pounds, although much of this was borne by the RAF. With a 1969 unit cost of 1.875m pounds 170 aircraft is a fleet cost 318.75m pounds, the development cost of 80-90m pounds gets to about 410m pounds leaving ~90m pounds for initial support costs.
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