formulation of testing gravitational redshift based on ...€¦ · presented ndat 2 issi workshop...

Presentedat2ndISSIWorkshop“Space4meMetrology,ClocksandRela4vis4cGeodesy”，25-28March2019,Bern

Formulation of testing gravitational redshift based on frequency

links between ACES on board ISS and ground station

Wen-Bin Shen, Xiao Sun

Wuhan University School Of Geodesy And Geomatics

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Content

2

3

4

One-way frequency transfer

Introduction

Formulation for testing gravitational redshift

Discussion

2ndISSIWorkshop“Space4meMetrology,ClocksandRela4vis4cGeodesy”

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Fig1.Testing of gravitational redshift is one of the critical part of testing the Einstein equivalence principle (EEP, Will 2014), which has been confirmed by various physicists (Pound and Rebka 1959, 1960a, b, 1965; Hafele and Keating 1972; Alley 1979; Vessot et al. 1980; Turneaure et al. 1983; Krisher el al. 1993).

Gravitational redshift

Introduction


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Fig2. Will (2014)

The most precise direct experiment of testing gravitational redshift is GP-A experiment (~ 70×10-6)

Fig3. Vessot and Levine (1979)

Introduction


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Recently, experiments performed by two Galileo satellites show better results (~ 2×10-5) (Delva et al. 2018; Herrmann et al. 2018). ACES mission: expected to be ~ 2×10-6.

Fig4. Cacciapuoti et al. 2017

Introduction


1

ACES payloads

A hydrogen maser (SHM) with medium-term frequency stability; A cold cesium atoms (PHARAO) with long-term frequency stability; Frequency stability: 1×10−13τ−1/2, with accuracy 2×10−16.

Introduction

Fig5. Cacciapuoti et al. 2017


Introduction 1

Two-way MWLs and ELT

Fig6. Laurent et al. 2015

Time signals on board are compared to ground clocks using microwave two-way time-transfer system (MWL) operating in the Ku band as well as a laser time-transfer system ELT.


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Frequency comparison VS time comparison

There are already a few literatures addressing to test gravitational redshift based on time comparison (Cacciapuoti and Salomon 2009; Duchayne et al. 2009; Meynadier et al. 2018), but there are short of publications related to frequency comparison.

Introduction


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Frequency comparison:

Strengths

1.  Hard to measure the instant frequency; 2.  Model is to be established

1.  Need no synchronization between ground and space; 2.  Weaken the influence of phase ambiguity; 3.  Require less time accumulation

Weaknesses

Here we formulate model of frequency comparison based on two-way MWLs

Introduction


1

Fig7. Cacciapuoti and Salomon (2009)

Concerning ISS’ orbit altitude of 400 km, for each pass over a given ground station, the observation time period is limited to 300-400 s. We aim at the frequency precision better than 2×10-16 in 300s

Introduction

We consider a downlink from satellite A to the ground B. In a general relativistic framework, accurate to 1/c3, the proper frequency shift of the photon from A to B is expressed as (Blanchet et al. 2001)

( )

( )

2

2

2

2

112

112

AE A

B B

A ABE B

vU rc q

qvU rc

νν

⎡ ⎤− +⎢ ⎥

⎣ ⎦=⎡ ⎤

− +⎢ ⎥⎣ ⎦

(1)


2 One-way frequency transfer

( )

( )23 2

41

B BA B AB B AB

AB B E BB

A B AB

r r RGM rq

c c r r R

⋅+ ⋅ −

⋅= − −

+ −

r vN vN v

( )

( )23 2

41

A AA B AB A AB

AB A E AA

A B AB

r r RGM rq

c c r r R

⋅+ ⋅ +

⋅= − −

+ −

r vN vN v

where

(2)

(3)


2

Doppler effect Shapiro effect


( )

( )

( )

( )

( )

( )

23 2

23 2

2

2

2

2

4

4

112

112

1

1

A AA B AB A AB

E AA

A B AB

B BA B AB B AB

E BB

A B AB

AE B

relB

E A

AB B

dopAB A

r r RGM rGc r r R

r r RGM rGc r r R

vU rc

AvU r

c

cA

c

⋅⎧ + ⋅ +⎪⎪ =

+ −⎪⎪

⋅⎪ + ⋅ −⎪

=⎪+ −⎪

⎪ ⎡ ⎤⎨ − +⎢ ⎥⎪ ⎣ ⎦=⎪⎡ ⎤⎪ − +⎢ ⎥⎪ ⎣ ⎦⎪⋅⎪ −⎪

=⎪ ⋅⎪ −⎩

r vN v

r vN v

N v

N v

Set simplified notation


2

(4)


Fig8. One-way frequency transfer


2

( )Brel dop A B

A

A A G Gνν

= + −(5)’

Models (4)-(5) hold only in vacuum. In a real space with medium, we have

Here is Doppler frequency value with considering atmospheric contributions

Adop


then we have

( )Brel dop A B

A

A A G Gνν

= + − (5)

On the other hand, the non-relativistic Doppler frequency shift effect is expressed as (Bennett 1968)


2

1 cosB

dop B B B A A AA

nf ds n nt

αλ

∂⎛ ⎞Δ = − + ⋅ − ⋅⎜ ⎟∂⎝ ⎠∫ T v T v

In the case of an isotropic media, α=0 (Davies 1965)

Caused by the time varying index

Caused by the velocity of both ends

(6)



2

Due to the velocity of source, the wavelength λ is expressed as

0 1 A A Anc

λ λ⋅⎛ ⎞= −⎜ ⎟

⎝ ⎠

T v(7)

Then, with eqs. (6) and (7), the received relative frequency due to Doppler effect is expressed as

1 cos

1

BB B B

Adopdop

A A A

n n dsf f c tA nfc

α⋅ ∂

− −+ Δ ∂= =⋅

−

∫T v

T v(8)


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In atmosphere, electromagnetic waves undergo direction change or refractive bending during their propagations. Hence, the direction of a refracted ray at the space station differs slightly from the un-refracted line-of-sight direction (Millman & Arabadjis 1984), as Figure 9 shows.

Figure 9. Principle of ray refraction through atmosphere for GNSS satellite. (Hoque et al. 2008)



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The refraction angle can be determined by Snell’s law (Croft and Hoogasian, 1968; Hoque et al. 2008)

0 00arcsin sinj

j j

r nr n

δ δ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

(9)



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With ISS’ height of around 400 km (Cacciapuoti & Salomon 2009), the ACES-ground links signals propagations experience atmospheric influences, therefore we have (Shen et al. 2016)

( )1 22

40.3 1cos ( )B

A Li Lt

n d dds t ds M M dst cf dt c dt

α ρ∂

= − + +∂∫ ∫ ∫

( )1 22

1 40.3 1+ ( )1

B B B

dop Li LtA A A

nd dcA t ds M M dsn cf dt c dt

c

ρ

⋅−

= − +⋅

−∫ ∫

T v

T v

Then, eq. (8) could be rewritten as

(10)

(11)



2

Ionospheric refraction index is relevant to carrier frequency, while tropospheric refraction index is not relevant to it. Therefore, for all links f1, f2 and f3, the bending effects of troposphere are almost same, which makes it easy to calculate the tropospheric part.

For ionospheric part, to the order of f-2, we have (Hajj and Roman 1998; Hoque et al. 2008)

21 40.3 ennf

= −

where ne is the electron density per cubic meter

(12)


For ISS ACES mission, the planned frequency links are as follows:

The frequencies of these three links are denoted as f1, f2, f3 2ndISSIWorkshop“Space4meMetrology,ClocksandRela4vis4cGeodesy”

Formulation for testing gravitational redshift 3

p  Ku band uplink, with carrier frequency 13.475 GHz, and frequency shift is known afterwards;

p  Ku band downlink, with carrier frequency 14.70333 GHz；

p  S band downlink, with carrier frequency 2248 MHz.

Fig 10. three links of ACES mission


3

A Ku band uplink signal f1 is emitted from ground station (B) at coordinate time t1 and received by space station (A) at t2. Meanwhile, two downlink signals f2, f3 are emitted from A at times t3 and t5 respectively, and received by B at times t4 and t6 respectively. If we definea coordinate time interval by Tij=tj-ti, T23 and T35 will be synchronized to be zero in theory, but actually they may not be zero.

Fig 11. Meynadier et al. 2017



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For ACES links： f1=13.475 GHz, f2=14.70333 GHz and f3=2248 MHz, 、 The third link is of low frequency, which suffers much from ionospheric effects. If the value of T35 is extremely small (<1 us), the only different error between links 2 and 3 is the ionospheric effect.



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Combining the three observed frequencies, we may determine the gravitational potential difference between space station and ground station. Details are referred to Sun and Shen (2019 in preparation for submission)



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•  Various factors will influence the accuracy of the determined potential.

hIps://www.na4onalgeographic.org hIp://www.sanandreasfault.org


Fig 12. diagram of earth tide Fig 13. diagram of plate motions


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Based GRT, we determine gravitational difference (GD): By conventional method, we determine GD: ΔUCM

ΔUGR =UB −UA


Fig 14. diagram of gravitational redshift


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Then we have

If the Einstein’s GRT is correct, coefficient β should be zero.


z = ΔUCM / c2 = 1+β( )ΔUGR / c

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Discussion 4

It is promising to test gravitational redshift at a level of 3×10-6

by frequency comparison. To achieve that accuracy level, the parameters accuracies should be carefully controlled, which need further investigations. At present accuracy requirement, it is not necessary to consider Earth’s tidal effects and plate motion effect. Simulation experiments are in process.

Thank you for your attention!

Email:[email protected]

formulation of testing gravitational redshift based on ...€¦ · presented ndat 2 issi workshop...

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