What the Heck is a ZHR?
By Wes Stone

The ZHR (Zenithal Hourly Rate) is an estimate of the number of shower meteors that would be seen by a single observer, watching an unobstructed area of the sky for a period of one hour, with the shower's radiant at the zenith and a limiting magnitude of 6.5.  Raw observed rates obtained under various conditions can be compared by first converting each to a ZHR.

In practice, the ZHR is determined from an observation by correcting the observational parameters to the "ideal" conditions listed above.  This is done via the rather vulgar-looking formula: ZHR=(F*C*K*N)/T, where:

F is a correction for any clouds or obstructions in the field,
C is a correction from the observer's limiting magnitude (LM) to LM=6.5,
K is a correction for the radiant's elevation,
N is the actual number of shower meteors seen, and
T is the time spent observing in hours.

For example, suppose you watched the Geminids from 8:30pm until 9:30pm on December 14 and saw 30 Geminids.  You kept track of percentage of cloudiness and limiting magnitude during the watch.  Let's say that, on average, 10% of your field of view was covered by clouds and your average LM was 6.0.

To determine F, you divide 1 by the proportion of clear sky in your field:
F = 1/0.9 = 1.11

C depends on both the limiting magnitude and the brightness profile of the shower.  When the average meteor is faint, you'll miss proportionately more meteors by observing in poor conditions.  The brightness profile is represented by the shower's population index, r.  r may be calculated from a large observed sample of meteors, but is rather complicated.  r is often assumed to be 2.0 for an initial analysis.  An r of 2.0 would mean that twice as many meteors appear in each magnitude class.  In other words, if there are 5 meteors of magnitude 1, there will be 10 of magnitude 2 and 20 of magnitude 3 (although you won't see all of them and will miss more faint meteors than bright ones).  C is then determined by:
C = r ^(6.5-LM) = 2.0 ^(0.5) = 1.41

K is determined using the radiant altitude (A) at the midpoint of the watch.  A can be found using a planetarium program or mathematical methods.  For our sample watch, A~30 degrees.
K = 1/sin(A) = 1/0.5 = 2

This means that when the radiant elevation is at 30 degrees, you see only half of the meteors that you would see if the radiant were at the zenith.  This is why it is better to observe in the morning, when the radiant elevation is higher.

We now know all the variables in the ZHR formula:
ZHR = (F*C*K*N)/T = (1.11 * 1.41 * 2 * 30)/1 = 94

Typically, the ZHR is expressed with error margins, the error being given as +/- ZHR/sqrt(N). So, we would have ZHR = 94 +/- 17.  For detailed analysis, the results of many observers must be pooled, as any single ZHR is but a small sample.  More data from more observers means that both statistical uncertainty and individual variations have less of an effect on the final results.  In addition, examination of a large number of observations where each meteor's magnitude has been recorded may help to refine the value of the population index r.

As a second ZHR example, let's work backwards.  The predicted ZHR may be used to forecast what kind of rates you'll see.  Remember, though, that if the prediction is wrong your forecast will be wrong, too.  Inherent unpredictability adds to the lure of observing meteor showers.  Still, this exercise shows the dramatic effect of light pollution on observed rates.

Our goal is to look at the Geminid rates a Northern Oregon or Southern Washington observer might expect when the radiant is at its highest elevation (~77 degrees) on the morning of December 14 at around 2am PST (10:00 UT).  We'll assume that our observer watches for one hour, from 1:30-2:30am, and that the skies are cloud-free.  At this time, six hours after maximum, a ZHR ~80 is reasonable.  If the observer's LM is 6.5, C=1 and the ZHR formula looks like:
ZHR = 80 = (1*1*1.03*N)/1

Solving for N, we find that N=78, nearly the ZHR. If one can find skies with LM>6.5, it is even possible for observed rates to exceed the calculated ZHR when the radiant is high.  On the other hand, an observer with LM=5.5 (assuming r=2.0) would see only half as many meteors, 39. Dark skies make a big difference!

It must be reiterated that the ZHR is an estimate.  The most valuable data are the raw numbers of a meteor watch: the shower association and magnitude of each meteor, along with complete notes on observing time and sky conditions.  However, since the ZHR is so often referenced (and misunderstood), I felt an explanation was in order.

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