How can I estimate relative humidity ($e/e_s$) from temperature and dewpoint?

Question

Background

my goal is to estimate vapor pressure deficit ($vpd$) from Relative Humidity ($rh = \frac{e}{e_s}$):

$$vpd = \frac{100 - \frac{e}{e_s}}{100} e_s$$

(from Hartmann "Global Physical Climatology)

But it is surprisingly difficult to find values of climatological mean $rh$ (e.g. see motivating question at gis.SE. It turns out that WeatherUnderground provides values of dewpoint.

Problem

At first I thought it would be trivial to convert from dewpoint to $e$ or $rh$, but according to Lawrence 2005, there is no direct conversion, only a "highly accurate conversion" to get from water vapor pressure ($e$) to the dewpoint ($t_d$) (equation 7):

$$t_d = \frac{B (ln\frac{e}{C})}{A - ln (\frac{e}{C})}$$

Where $[A,B,C]$ are empirical constants$[17.625, 243.04^oC,610.94Pa]$

Questions

2. why is there no exact relationship between $e$ and $t_d$?
3. how can I solve the last equation for $e = f(t_d)$ (or $rh$)?

Answer 1

Still working on the answer to question 1, but in theory the water vapor pressure is the partial pressure of water such that no more water will evaporate or condense (ie. the air is saturated) and is a function of temperature and pressure (see this for example). The line is the saturation point. So strictly speaking, there is not an exact relation between vapor pressure and temperature at saturation unless a pressure is assumed. For atmospheric work, they may just assume 1 atmosphere always and come up with the empirical relations that are usually close enough.

For the second question, you can move $t_d$ to the right hand side so it is set to zero and use Newton's Method or the Secant Method to find the root. Since it is analytical, you actually can find the derivative function needed to make it easier.

For a totally unrelated purpose, I made a spreadsheet that does the calculations between partial pressure/dew point/relative humidity. You can check it out at http://www.dicklanevelodrome.com/topic/2010/06/physics-and-track-part-2.html and see how the internals work.