Math Pages

• One in sixty rule
• Reciprocals
• Get down

Estimating ETA

• Chew on This
• Sixes and sixties
• Two-bit math
• It's in the percentages
• Leading a DME arc

Fun Stuff

• Change?
• Ace on top
• Ferry Pilots
• Pilots and Mechanics
• Add Fast

Intercepting a DME Arc

DME Arcs are a common part of modern IFR approaches. They allow the pilot to bypass the procedure turn an thus save time even when radar service is not available. When radar service is available they allow pilot navigation with reduced controller workload, thus increasing the number of aircraft that can be handled at a some airports.

	In the picture to the left the back circle represents a DME arc (there would be VOR at the center of this circle. The red line is a radius line. The green line is tangent to the circle, which means that it is at a right angle to the radius. An aircraft could be approaching the arc from either of the two directions shown. On this page I will develop a formula that the pilot can use to judge when to start the turn so that the airplane will wind up exactly on the arc.
	Let us consider first the case of approaching the arc from within, as in the picture to the left. The blue circle represents the radius of turn of the airplane. This circle depends on the speed of the airplane, and the angle of bank. We will call the radius of this turn "r." If the airplane begins to turn a distance r back from the arc the airplane will roll out not on the arc, but on the tangent line. In other words the airplane will overshoot the arc. I will not provide a diagram, but you should be able to see that the airplane approaching the arc from outside would undershoot the arc slightly. I.E. in the upper diagram both airplanes will wind up on the green tangent line, not the actual arc.
	Radius of Turn Formula First we must commit to using a RATE ONE TURN. By definition a rate one turn takes two minutes to perform a 360° turn. The circumference, or distance around a circle is given by two pi r, as shown in the picture to the left. Therefore, the radius of turn (in NM.) is TAS x pi/60, or about 0.005 x TAS. This is usually stated as half of one percent of TAS. For example if you are flying 100 knots one percent is 1Nm, so the radius of turn is .5NM.

A Common Mistake

The most common mistake beginners make is to start rolling into the turn to intercept an arc at .005 x TAS. You MUST roll in before that. There are two reasons for this:

1. More lead is required when intercepting from within the arc (less from outside.)
2. The turn must be established at .005 x TAS

When you roll into a turn it takes a few seconds from the time you begin to roll until the airplane is actually established in the turn. I personally am a big fan of passenger comfort considerations, and that means a nice slow roll rate. At the speed of most light twins that means about six seconds from the time you start to turn the control wheel until the turning is established at rate one (in slower airplanes it happens faster, in jets it takes longer.) Using 120 knots as an example speed the airplane travels 0.2NM in six seconds. So, when intercepting from within an arc you need to add about .3 NM to the formula (see the diagram to understand the extra 0.1.) If intercepting from outside the arc you may only need to add 0.1NM.

The Formula For Arc Intercepts

The IPM suggests half of one percent of the TAS (or groundspeed) when intercepting arcs. We can now see that the people who wrote this did some math, but they should have gone a bit further. Our formula is:

Lead Arc by - .5% of TAS + 0.1 to 0.3 (as required.) Knowing just how much extra is "required" is part of being a professional. Examine the diagrams above to make sure you know all the variables, then do a little practice. You will find that since you fly the same airplane every day you will have the same lead every day - so you don't really do math in flight anyway, you just lead by the amount you have discovered works, which as you now can see, depends to in part on how quickly you roll into turns and whether or not you accurately do rate one turns.

The other thing to keep in mind is that this formula is ONLY for ZERO WIND. The airplane actually drifts with the wind while you are making the turn. The turn takes approximately 30 seconds, so you will drift by windspeed/120 Nm. during the turn. In other words a 12 knot wind will drift you 0.1Nm. in 30 seconds, a 24 knot wind will drift you 0.2Nm. and so on. The wind drifts you in the direction that the wind is blowing (which an IFR pilot should always know.) You should allow for the wind and increase or decrease your lead to the arc accordingly.

Final Note

The analysis on this page is ONLY for the case where the arc is intercepted at 90°. This is quite common, but by no means the only possibility. If you are intercepting at a shallower angle, as shown in the picture to the left then your lead will be much less than calculated here.