1. Relativity Models in VLBI
John Gipson
NVI,Inc/NASA GSFC
2017-July-10
Unified Analysis Workshop
University Paris-Diderot
Paris, France
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2. Motivation Question: What goes into Relativity Models?
Could they be responsible for the VLBI-SLR scale-difference?
Compare
1990 ‘Consensus Model’ of T. Eubanks, et. al. (current IERS standards)
2016 ‘Advanced relativistic VLBI model for geodesy’, M. Soffel,
S. Kopeiken, and W-B. Han.
Other effects?
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

3. Searching for Scale Differences
𝜏 12 = 𝐵 12 ∙ 𝑘 𝑐
Our observable is the differential delay:
This is an idealized version: In a vacuum, and ignoring both special and general relativity.
In practice lots of correction terms.
Suppose that instead of the above I measure:
𝐵 12 ∙ 𝑘 𝑐 1+𝛼
Then the station positions will be magnified by the ( ). Want α ~0.7e-9
Stated another way, want to be alert to anything that multiplies the delay.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

4. The Models Consensus Model (CM) Advanced Model (AM)
Don’t look very similar!
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

5. Dictionary (for consensus model)
Each model has it’s own
dictionary. Symbols mean different things in the two models, and you must be very careful when comparing the two.
E.g. ‘w’
CM: geocentric site velocity AM: gravitational potential
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

6. Expansion in 1/c
To compare CM and AM models, will expand CM in powers of 1/c.
𝑏 𝑚𝑎𝑥 𝑐 = 12,000𝑘𝑚 300,000𝑘𝑚/𝑠 =4× 𝑝𝑠
𝑉 𝐸𝑎𝑟𝑡ℎ 𝑐 = 30 𝑘𝑚/𝑠 300,000 𝑘𝑚/𝑠 = 10 −4
𝑤 𝑠𝑖𝑡𝑒 𝑐 = 0.46 𝑘𝑚/𝑠 300,000 𝑘𝑚/𝑠 = 1.5×10 −6
To be good at the ps level, only need to keep terms up to 1 c 3
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

7. The Models Sometimes I expand the denominator in 1/c
Gravitational effects
We will come back to gravitational part soon.
Atmosphere handled separately in CM but equivalent to this.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

8. The Models Terms like 𝑉 𝐸 𝐵
The 1/c^3 term includes contributions from the numerator and expanding the denominator
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

9. The Models Terms like 𝑘 𝐵
1/c^2 and 1/c^3 come from
expanding denominator in a power series.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

10. The Models Terms like 𝑘 𝐵
Last three terms in the consensus model. You can ignore denominator if you are keeping terms ~1/c^3.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

11. The Models Advanced Model includes 3 additional terms.
First two are related to acceleration.
Last term is ~ few fs.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

12. Advanced Model Acceleration terms
𝐵∙𝑘 𝑐 𝑎 𝑒 ∙ 𝑋 2 𝑐 2 = 𝐵∙𝑘 𝑐 𝑉 𝑒 2 𝑅 𝑒 𝑋 2 𝑐 2 = 𝐵∙𝑘 𝑐 𝑉 𝑒 2 𝑐 2 𝑋 2 𝑅 𝑒
Term in () ‘sort-of’ looks like scale factor….
𝑎 𝑒 ∙ 𝑋 2 𝑐 2 = 𝑉 𝑒 2 𝑅 𝑒 𝑋 2 𝑐 2 = 𝑉 𝑒 2 𝑐 2 𝑋 2 𝑅 𝑒 = 10 −8 6× × =0.4× 10 −12
But we are looking for something like 10^-9.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

13. Gravitational Effects
The Consensus model includes additional terms like:
Where the vectors are from the receiver to the gravitating body.
For the Earth this simplifies to:
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

14. Gravity Effects
Because of GR effects, the travel time of light is greater than it is in Euclidean space.
This effect was first proposed by Shapiro in 1964, and verified in using the MIT Haystack Radar which measured the round-trip travel time of radar reflected from Mercury and Venus.
Light ray
𝑟 0 𝑟
The explicit formula to O(G) is given by:
Weinberg: Gravitation and Cosmology
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

15. Application to Earth. A Path 1 B 𝜃 1 𝑅 1𝐴 =(𝐿 𝑅 𝐸 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑅 1𝐴 =(𝐿 𝑅 𝐸 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑅 1𝐵 =( 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
Consider a radio wave coming on Path AB. We start with finite L and later let it go to infinity.
Weinberg’s formula gives the time delay to the vertical line. We want to subtract off the dashed line.
𝑡 𝑅 𝐴 − 𝑅 𝐵 =𝑡 𝑅 𝐴 − 𝑅 0 −𝑡 𝑅 𝐵 − 𝑅 0
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

16. Application to Earth: First term.
𝑅 1𝐴 =(𝐿 𝑅 𝐸 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑅 1𝐵 =( 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑡 𝑅 𝐴 − 𝑅 𝐵 =𝑡 𝑅 𝐴 − 𝑅 0 −𝑡 𝑅 𝐵 − 𝑅 0
𝑐 𝑡 1 = 𝑅 1𝐴 2 − 𝑅 𝐸 2 𝑠𝑖𝑛 2 𝜃 𝑅 1𝐵 2 − 𝑅 𝐸 2 𝑠𝑖𝑛 2 𝜃 2 =𝑐 𝑅 𝐸 (𝐿−𝑐𝑜𝑠 𝜃 1 )
Differential delay:
Note that L drops out.
𝑐 𝑡 2 −𝑐 𝑡 1 = 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 − 𝑅 𝐸 𝑐𝑜𝑠 𝜃 2 = 𝑘∙ 𝐵 21
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

17. Application to Earth: Second term.
𝑅 1𝐴 =(𝐿 𝑅 𝐸 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑅 1𝐵 =( 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑡 𝑅 𝐴 − 𝑅 𝐵 =𝑡 𝑅 𝐴 − 𝑅 0 −𝑡 𝑅 𝐵 − 𝑅 0
𝑐 𝑡 1 =2 𝐺𝑀 𝑐 2 𝑙𝑛 𝐿 𝑅 𝐸 + 𝑅 𝐸 𝐿 2 + 𝑠𝑖𝑛 2 𝜃 2 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 −2 𝐺𝑀 𝑐 2 𝑙𝑛 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 + 𝑅 𝐸 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1
Differential delay:
𝑐 𝑡 2 −𝑐 𝑡 1 =−2 𝐺𝑀 𝑐 2 𝑙𝑛 𝐿 𝑅 𝐸 + 𝑅 𝐸 𝐿 2 + 𝑠𝑖𝑛 2 𝜃 1 𝐿 𝑅 𝐸 + 𝑅 𝐸 𝐿 2 + 𝑠𝑖𝑛 2 𝜃 𝐺𝑀 𝑐 2 𝑙𝑛 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 + 𝑅 𝐸 𝑅 𝐸 𝑐𝑜𝑠 𝜃 2 + 𝑅 𝐸
This vanishes for large L
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

18. Application to Earth: Second term.
𝑅 1𝐴 =(𝐿 𝑅 𝐸 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑅 1𝐵 =( 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑡 𝑅 𝐴 − 𝑅 𝐵 =𝑡 𝑅 𝐴 − 𝑅 0 −𝑡 𝑅 𝐵 − 𝑅 0
𝑐 𝑡 1 =2 𝐺𝑀 𝑐 2 𝑙𝑛 𝐿 𝑅 𝐸 + 𝑅 𝐸 𝐿 2 + 𝑠𝑖𝑛 2 𝜃 2 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 −2 𝐺𝑀 𝑐 2 𝑙𝑛 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 + 𝑅 𝐸 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1
Differential delay:
𝑐 𝑡 2 −𝑐 𝑡 1 =+2 𝐺𝑀 𝑐 2 𝑙𝑛 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 + 𝑅 𝐸 𝑅 𝐸 𝑐𝑜𝑠 𝜃 2 + 𝑅 𝐸
This is in the consensus model gravitational delay.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

19. Application to Earth: Third term.
𝑅 1𝐴 =(𝐿 𝑅 𝐸 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑅 1𝐵 =( 𝑅 𝐸 𝑐𝑜𝑠 𝜃 1 , 𝑅 𝐸 𝑠𝑖𝑛 𝜃 1 )
𝑡 𝑅 𝐴 − 𝑅 𝐵 =𝑡 𝑅 𝐴 − 𝑅 0 −𝑡 𝑅 𝐵 − 𝑅 0
𝑟− 𝑟 0 𝑟+ 𝑟 0 = 𝑟 2 − 𝑟 𝑟+ 𝑟 = 𝑥 2 𝑟+ 𝑟 0 2
Useful identity
𝑐 𝑡 1 = 𝐺𝑀 𝑐 2 𝐿 𝐿 2 + 𝑠𝑖𝑛 2 𝜃 1 +𝑠𝑖𝑛 𝜃 1 − 𝐺𝑀 𝑐 2 𝑐𝑜𝑠 𝜃 1 1+𝑠𝑖𝑛 𝜃 1 -
𝑐 𝑡 2 −𝑐 𝑡 1 = 𝐺𝑀 𝑐 2 𝑐𝑜𝑠 𝜃 1 1+𝑠𝑖𝑛 𝜃 1 − 𝑐𝑜𝑠 𝜃 2 1+𝑠𝑖𝑛 𝜃 2
Not in consensus model
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

20. Third term is an effective scale factor.
𝑐 𝑡 2 −𝑐 𝑡 1 = 𝐺𝑀 𝑐 2 𝑐𝑜𝑠 𝜃 1 1+𝑠𝑖𝑛 𝜃 1 − 𝑐𝑜𝑠 𝜃 2 1+𝑠𝑖𝑛 𝜃 2
Plot
x=𝑐𝑜𝑠 𝜃 1 −𝑐𝑜𝑠 𝜃 2
y= 𝑐𝑜𝑠 𝜃 1 1+𝑠𝑖𝑛 𝜃 1 − 𝑐𝑜𝑠 𝜃 2 1+𝑠𝑖𝑛 𝜃 2
500 random pairs of cosines.
Y almost looks like baseline, with some scatter.
Slope=0.73
Scale factor
𝛼=0.73× 0.695e-9=0.5e-9
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France

21. Conclusions Consensus Model and Advanced Model agree at the ps level.
Possible explanation for some of the scale difference.
John Gipson NVI, Inc./NASA GSFC
2017-July-10
Unified Analysis Workshop, Paris, France