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The equations can be found in the IC's datasheet, and the bit rate value is valid for a 255 byte packet (one SSDV packet for example).
Since this is not actual bit rate, but a value derived from time on air of such packet divided by the size of only the payload data, the effective bit rate decreases with less data sent by the user.
From the bottom up: the slowest Mode 0, then Mode 1, all the way to a very fast and quite faint Mode 9. Comparing Semtech's and Hope RF's datasheets, it seems that the modules simply use the Semtech's SX127x ICs while mainly providing the external filtering and impedance matching for a specific frequency band.
I also designed simple adapter boards which now host my modules.
The signal level decreases as the energy is spread across a wider bandwidth.
To better imagine the difference between individual modes, in other words the parameters, this is a screen grab from SDRsharp's waterfall again capturing the same 57 byte packet transmitted by all ten modes in succession with 250ms gaps in between transmissions. The modules I opted for were Hope RF's RFM96 intended for the 434MHz band, ordered on Ebay.
The signal is comprised of a series of up-chirps and down-chirps all within the selected bandwidth.
Data is then represented by instantaneous changes in frequency between individual chirps.
All of this including the whole demodulation and decoding process is brilliantly described in this presentation by Matt Knight.
I added Mode 9 on top of that for testing the highest data rate configuration Lo Ra provides.
The table above summarizes the parameters of individual modes and provides a calculation of an effective bit rate.