Re: [Phys-L] Doppler

[Todd Pedlar <pedlto01@luther.edu>]

So I've not looked up the original papers, but the shift for a
non-relativistic velocity from which radar waves are reflected is twice the
ratio of the object speed to that of light, times the original frequency.
So the frequency shift for 3 m/s on the limb of Mercury would result in
about a 48 Hz shift in the original 2.4 GHz emission frequency.  I imagine
that since Mercury is moving, and so all the light is shifted in an overall
sense because of that much larger speed, they'd have to have compared the
range of frequencies received and compare the directly reflected light
(from the center) to that maximally shifted light (from the limbs) and
subtract.

The exercise of showing students how the shift of 2*f*v_obj/c would be a
good exercise for them, though, starting from the shift of
frequency received by the limb, f_limb = f_sent * sqrt((1-beta)/1+beta)),
and then shifting again because the reflected waves are being
reflected from a moving surface, f_received = f_limb *
sqrt((1-beta)/(1+beta)) = f_s * (1-beta)/(1+beta).

Then the overall shift would be f_sent - f_received = f_s * (1 -
(1-beta)/(1+beta)) = f_s * 2 * beta / (1+beta), but since beta is 10^-8,
this is about 2*f_s*beta.  With a wavelength of 12.5cm sent to Mercury,
which corresponds to a frequency of 2.4 GHz, a frequency shift of 48 Hz
would be expected if the rotation speed is 48 Hz.

> From that 48 Hz shift (relative shift of 20 x 10^-9) would mean a shift in
wavelength of 2.5 nm.

So... yeah, tiny :)

T

On Tue, May 27, 2025 at 9:58 AM Anthony Lapinski via Phys-l <
phys-l@mail.phys-l.org> wrote:

> Does anyone know how to calculate Mercury's rotation rate? I mean, Doppler
> data was taken in the 1960s to determine its period to be around 59 days.
> This corresponds to an equatorial speed of only 3 m/s. I think a radio
> telescope was used, with a wavelength of 12.5 cm. What is the corresponding
> wavelength shift? I imagine very tiny! No calculations found online for
> this, just the rotational period result. Wanted to make this a problem for
> my HS astronomy students, but 3 m/s << speed of light in the Doppler
> formula.
> _______________________________________________
> Forum for Physics Educators
> Phys-l@mail.phys-l.org
> https://www.phys-l.org/mailman/listinfo/phys-l


-- 
Todd K. Pedlar
Professor of Physics and Physics Department Head
Luther College, Decorah, IA
pedlto01@luther.edu
(563) 387-1628
*Learner | Context | Strategic | Individualization | Achiever*