Is there a quantifiable tradeoff (in terms of effective broadcast distance) between the height of a radio tower and the power of the transmitter?

I have a dataset on radio stations with the following variables:

• 
  

  the power of the transmitter, in watts

  
  


• 
  

  the coordinates of the radio tower

  
  

I can easily get the elevation at those coordinates and the average elevation of the surrounding area. I know that estimating the broadcast range or the radio horizon of a transmitter is a complicated question, and I know there are simple estimates like this:

\begin{equation} \text{horizon}_{\text{km}} \approx 3.57 \sqrt{\text{height}_{\text{metres}}} \end{equation}

Are there similarly simple estimates for how this horizon changes with the power of the transmitter, assuming the same frequency, elevation, and tower height?

For example, if there are two towers of the same height at the same coordinates (yes, I know this is impossible), but one has a 1 kW transmitter and the other has a 50 kW transmitter, is there a way to adjust the equation above to account for this difference in power? Am I wrong to assume that the more powerful transmitter will have a larger range, all else being equal?

Answer

The power of the transmitter is completely independent of the height of the tower: it depends only on some design features of the antenna: the actual shape, the input current, the impedence etc. Based on geometrical arguments, the power roughly drops like $1/r^2$ (at least in the far field).

The angular distribution of this power does depend on antenna characteristics, i.e. a half-wave dipole does not radiate in the same way as a Hertzian dipole. This will depend on the relative geometry of the antenna and the reception point, v.g. there might be particular orientations where the antenna doesn't radiate for instance.

As specific example, the radiation pattern of a short (or Hertizian) dipole looks like

i.e. the short dipole does not radiate in the line of the antenna. Intuitively, this can be understood as there is no magnetic field in the line along a finite current-carrying wire that generates the magnetic field.

For the full-wavelength dipole we have a more flatten pattern

and, for some specific antenna arrays, one can obtain very directional signals, as shown below:

As a result, concentration of the signal (as measured by the directivity of the antenna), is highly dependent on the type of antenna. Clearly the directivity will affect the range, and the reception of the signal will depend on the relative orientation of the receiver w/r to the antenna.