# How to estimate the time to intercept for homing missiles?

I have been reading the book Tactical and Strategic Missile Guidance by Zarchan, and a notice that he frequently uses the time to intercept $$t_{go}$$, and in his simulations he fixes the final time.

My question is if $$t_{go}$$ also is used in practice, and if it is not the case what is done instead?

The reason for me asking this is that I have doubts about the practicality of estimating $$t_{go}$$ as it could change dramatically if the target does an unexpected manoeuvre. Additionally estimating $$t_{go}$$ requires that we compute a predicted impact point which seems cumbersome in an endgame scenario compared to proportional navigation on the form $$a_c = N V_c \dot{\lambda}$$. Any references to how $$t_{go}$$ can be computed is also appreciated.

• Boringly I suspect the answer to this is not going to be found on the public parts of the internet, however assuming steady course and speed is right for a large number of cases (drone/unaware human), and attempts to predict likely target response would be very prone to gaming/countermeasures. Jan 23, 2022 at 12:36
• That makes sense @GremlinWranger I know from experience that certain missile has a lot of stuff going on behind the scenes, but I was afraid I was completely overlooking something basic Jan 24, 2022 at 13:56
• @pedernv perhaps I should add to my answer that I frequenly saw this value used in missile simulation, although in missile guidance and control it is still used. Nevertheless the concept of calculation is exactly as I laid out below. Aug 11, 2022 at 16:54
• @pedernv No Mr ! There is an explanation to your problem regarding this matter and it is understandable and valid. Yes, in practice things tend to go technical rather than theoretical. Please read my answer below.
– user64784
Aug 13, 2022 at 11:56
• Could you re-phrase 'if 𝑡𝑔𝑜 also is used in practice, and if it is not the case what is done instead'? That might be perfectly expressed in a dedicated field of maths, but how could it work in general English? Sep 29, 2022 at 17:45

## General

Knowing the time-to-go ($$t_{go}$$) is essential for advanced missile systems, simply because it defines which phase of the intercept the missile is in. This helps to inform logic and guidance decisions as well as how aggressive they are, which in turn shall maximize the probability of intercept compared to simpler algorithms such as proportional guidance.

## Application examples

Such logic decisions may be for example when to switch on the missiles active radar system, which in turn might alert your enemy of your presence, which motivates to do that as late as possible. The AMRAAM missile uses such a system. Other missiles change their flight behaviour for example the MBDA Meteor throttles up and down to optimize its energy when reaching its target and even changes its control strategy (bank-to-turn for skid-to-turn) when it is about to hit its target even though that will starve its ramjet engine. Another example is that of a so-called "loft" whereby long-range missiles loft their trajectory to fly through upper parts of the atmosphere in order to take advantage of lower air-drag. Again this kind of behaviour cannot be realized without the compution of your targeted impact point and therefore an explicit knowledge of $$t_{go}$$.

## Comparison to proportional guidance

2. You imply that the proportional navigation would be the better solution which would negate the need for the computation of $$t_{go}$$. As given by the examples above, $$t_{go}$$ can be used to increase the chance of hitting a target. Additionally proportional navigation is not perfect either. For example, if you employ PN against a target very far away which only maneuvers slightly (for example by executing a very minor sinusoid motion) your target point will move very dramatically. This results in great maneuvering on the side of the interceptor missile as you try to keep up with this target point. This excessive maneuvers leads to the missile bleeding off a lot of its energy. Allegedly, this happend with Patriot missile defending against Iraqi modified scud missiles which became unstable in the air, however I cannot find the source in the moment. The estimation of $$t_{go}$$ could inform a decision to adapt your guidance behaviour in order to rule out premature energy loss in midcourse flight.
### How to compute $$t_{go}$$
It is as simple as y dividing your velocity by remaining distance $$t_{go} = \frac{x_{go}}{V_{missile}}$$. The trick is to know these to parameters over your flight. Modern missile such as the AMRAAM explicitly calculate their trajectory over their flight path until the predicted point of intercept and therefore know exactly how far they intend to fly. From this they can then explicitly calculate how long they will fly. This is computationally expensive, however as stated before modern missile do contain the necessary computing power to carry out these kinds of calculations and they use it! For example look at the different flight profiles of a Javelin missile or read around a bit on the different modes and functionalities of modern missiles, you will be amazed of what kind of complex designs and logic trees these include.