How did NASA figure out when and where the Apollo capsule would touch down on the ocean?

Question

After watching the amazing footage of the Apollo 13 capsule parachuting down on the ocean, I was intrigued by how NASA predicted when and where it would splashdown on the vast ocean's surface with amazing precision. The time and location had to be predicted far in advance, considering they had to account for the travel time of the slow moving recovery vessel. This is especially impressive considering the capsule was in free-fall without any guidance system, leaving its eventual landing location at the mercy of the elements, such as wind and turbulence.

Answer 1

At high altitude, the effects of the air on the falling capsule are minimal. The main parachutes don't deploy until around 2 miles altitude, and the capsule is still coming down at about 22 miles per hour at splashdown, so it's easy to see that even in very high winds it couldn't be blown off course by more than a small number of miles.

The spacecraft doesn't maneuver much on the return trip from the moon; once its final midcourse correction is made (two days before splashdown, for Apollo 11) the trajectory can be projected fairly accurately (although, for Apollo 13, venting from the LM's thermal management system did push it off course, and they had to adjust their trajectory estimates as they went).

That leaves maneuvering during the reentry phase as the primary source of uncertainty in the location of landing. Apollo flies a lifting trajectory once it begins to enter the atmosphere, in order to keep from descending too steeply into dense air and suffering a high-g deceleration. In this phase, the capsule, under manual control or automatic guidance, rolls to adjust the amount of lift -- if the capsule is oriented vertically in the roll plane, it gets maximum lift, "going long"; if the rolled towards the horizontal, it descends more steeply (while pulling slightly to the right or left). Depending on slight variations in how steep the approach starts, and how it flies this phase, there can be a fair amount of variation in where it ends up.

According to Apollo By The Numbers, the crewed Apollo flights all splashed down within 3 nautical miles of the target point. Apollo 14 hit closest, at 0.6 miles, Apollo 16 furthest at 3.0.

Answer 2

I am not sure that the existing answers really address the question asked, namely

How did NASA figure out when and where the Apollo capsule would touch down on the ocean?

The short answer is that NASA targeted a landing point and set up the entry guidance software to fly to that point.

They knew where it was going to land because they sent it there!

For the details of how they targeted it, see Apollo experience report: Mission planning for Apollo entry. Read the chapter titled ENTRY TARGETING. It starts out

Two types of targeting are used for the Apollo entry: velocity and flight-path angle V, y targets at the entry interface and entry range. The V, y targets are selected to ensure aerodynamic capture by the atmosphere of the earth while entry-ranging capability and acceptable aerodynamic heating conditions are maintained. The target entry range is chosen to be compatible with GNCS performance and to enhance the entry monitoring and backup control capabilities.

The entry-ranging capability is not used for control of the landing position relative to the surface of the earth. That is, the landing latitude is essentially at the lunar antipode at the time the CM enters the sphere of influence of the earth. This restriction occurs because the transearth trajectory must pass over the lunar antipode once the gravitational potential of the earth becomes the predominant force field, and the relatively short entry ranges result in the landing point always being located near the antipode. The landing longitude is controlled by varying the transearth injection time and by varying the transearth transit time to permit the earth to rotate to a favorable position relative to the transearth and entry trajectories. Therefore, the primary use of the variable entry-ranging capability is to make relatively small adjustments to the planned landing point during the mission to avoid bad weather conditions that may develop in the landing area.

Once the targets were set up and the vehicle was at Entry Interface, the automatic guidance system flew the capsule to the landing site.

From the Apollo 11 Crew Technical Debrief:

We gave spacecraft control over to the computer after we passed all our pitch attitude cross-checks. We gave it to the computer shortly before 400,000 feet. I don't recall exactly when, but a matter of seconds before 400,000 feet. We stayed in CMC, AUTO for the rest of the entry. The computer did its usual brilliant job at steering. We just sort of peered over its shoulder and I made sure that the spacecraft was responding to the bank angles that the computer commanded, and that those bank angles made sense in light of what we saw on the EMS and through other bits and pieces of information.

Acronymology:

• CM: Command Module
• CMC: CM Computer
• EMS: Entry Monitor System
• GNCS: Guidance, Navigation, and Control System

Answer 3

As answered here, all of the Apollo missions had accurate splashdowns. This level of accuracy actually started with Project Gemini several years earlier. The Gemini capsule was the first spacecraft which had the ability to maneuver to this level of precision during reentry.

The first three Gemini missions had various problems which led to splashdowns that were well off target. Reminiscent of some of the embarrassing mistakes seen in the past few years in spaceflight, during the Gemini 5 reentry the computers worked perfectly, however someone had inadvertently entered into the flight software the Earth's rotation as 360 degrees per 24 hours instead of 360.98, causing the capsule to miss the splashdown target by over 100 kilometers.

However the next several flights had increasingly accurate splashdowns, with the most accurate of the Gemini program being the splashdown of Gemini 9A less than a kilometer from its target.

Mission	Date	Kilometers	Miles	Nautical Miles
Gemini 3	3/23/1965	84	52	45
Gemini 4	6/7/1965	80	50	43
Gemini 5	8/29/1965	129	80	70
Gemini 6A	12/15/1965	18	11	10
Gemini 7	12/18/1965	12	7	6
Gemini 8	3/17/1966	2	1.2	1.1
Gemini 9A	6/6/1966	0.7	0.43	0.38
Gemini 10	7/21/1966	5.6	3.5	3
Gemini 11	9/15/1966	4.5	2.8	2.4
Gemini 12	11/15/1966	4.8	3	2.6

Gemini 9A splashdown (NASA)